Supersymmetric Integrable Hamiltonian Systems, Conformal Lie Superalgebras $\mathcal{K}$(1, N = 1, 2, 3), and Their Factorized Semi-Supersymmetric Generalizations

Abstract

We successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable super-Hamiltonian systems on functional $N=1,2,3$- supermanifolds, possessing rich supersymmetries and endowed with suitably related compatible Poisson structures. As an application, we describe countable hierarchies of new nonlinear Lax-type integrable $N=2,3$-semi-supersymmetric dynamical systems and constructed their central extended superconformal Lie superalgebra $\mathcal{K}(1|3)$ and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals. Moreover, we generalized these results subject to the suitably factorized super-pseudo-differential Lax-type representations and present the related super-Poisson brackets and compatible suitably factorized Hamiltonian superflows. As an interesting point, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of the centrally extended super-pseudo-differential operator Lie superalgebras on the $N=1,2,3$-supercircle.

1. Introduction

One of the most important legacies of modern mathematical physics is deemed nowadays as a new fruitful conception of supersymmetry, whose main concept [1,2] is to treat elementary bosonic and fermionic particles equally, which is what mathematically amounts to incorporating anticommuting Grassmann-type variables together with the usual commuting variables. In such a way, a number of well-known mathematical physics equations have been generalized into supersymmetric analogues, among which we find [3,4,5,6,7] supersymmetric versions of sine-Gordon, Korteweg–de Vries, Burgers, Kadomtsev–Petviashvili hierarchy, Boussinesq, the nonlinear Schrödinger equation, and many others. The investigation of the problem of particle-like behavior in supersymmetric field theories naturally leads to a theory of super-integrable systems and studying their properties, which can be helpful in part in the analysis of modern super-string [8] mathematical physics problems. It is nowadays well known [9,10,11,12,13,14] that there exist integrable fermionic extensions [7] of the completely integrable field theory systems on functional supermanifolds, related to conformal superalgebra symmetries, and which are not supersymmetric. It was also observed [5,15,16,17,18,19,20] that, in most cases, the supersymmetric integrable extensions were associated [7,21,22,23] to evolution superflows, generated by means of super-Lax-type representations, yet, as was clearly demonstrated in the work [24], there exist also supersymmetric integrable extensions, related to semi-supersymmetric systems, generated by the centrally extended $N=2$-superconformal loop Lie superalgebra symmetry and allowing for a reduction to supersymmetric flows. Here, it is also worth mentioning that, in fact, almost all of the so-called “new” nonlinear $N=1$-super-integrable dynamical systems, published [3,4,11,25,26] during the past few decades, are related to coadjoint flows of the affine conformal $N=1$-symmetry Lie superalgebra $\mathcal{K}(1|1)$ described in detail first in [14], and, in general, are related [27] to the affine Sturm–Liouville-type superconformal spectral problem $(-D_{\theta }^3+({\sum }_{j=-m}^{m+p-1}(u_j\theta +v_j){\lambda }^j)+\theta {\lambda }^{m+p})f(x,\theta )=0,m,p\in \mathbb{N},\lambda \in \mathbb{C},$ on the supercircle ${\mathbb{S}}^{1|1}\simeq \{(x,\theta )\in {\mathbb{S}}^1\times {\mathsf{\Lambda }}_1^{(1)}\}$ for a smooth function $f\in C^{\infty }({\mathbb{S}}^{1|1};{\mathsf{\Lambda }}_0^{(1)})$, where ${\mathsf{\Lambda }}_0^{(1)}\oplus {\mathsf{\Lambda }}_1^{(1)}$ $:={\mathsf{\Lambda }}^{(1)}$—the corresponding ${\mathbb{Z}}_2$-graded superalgebra. Meanwhile, as the description of $N=1$-supersymmetric Lax-type flows [5,6,7,9,19,20,21,28,29] is known widely enough, the $N\ge 2$-supersymmetric integrable flows, before being analyzed in [6,15,16,22,30,31], still should be paid more attention.

Owing to the interesting observation in the work [24], based on the affine Sturm–Liouville-type superconformal spectral problem $(D_{{\theta }_1}{D}_{{\theta }_2}+{\sum }_{j=-m}^{m+p-1}{u}_j(x,\theta ){\lambda }^j+$ ${\lambda }^{m+p})f(x,\theta )$ = 0, $m,p\in \mathbb{N},\lambda \in \mathbb{C},$ on the supercircle ${\mathbb{S}}^{1|2}\simeq \{(x,\theta )\in {\mathbb{S}}^1\times {\mathsf{\Lambda }}_1^{(2)}\},$ the special reductions in the related nonlinear integrable superconformal evolution flows prove to be supersymmetric dynamical systems on the corresponding functional supermanifolds. An interesting Backlund-type construction of nonlinear $N=2$-superconformal semi-supersymmetric dynamical systems was suggested within the Lie-algebraic approach in [12], generalizing in part those obtained before in [15].

In the present work, we successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable-in-general semi-supersymmetric Hamiltonian systems on functional $N\ge 2$-supermanifolds, possessing rich yet hidden supersymmetries and endowed with suitably related super-Poisson structures. As an application, we describe countable hierarchies of new Lax-type integrable nonlinear $N=3$-semi-supersymmetric dynamical systems. In particular, we analyze the suitably central extended superconformal affine Lie superalgebra $\widehat{\mathcal{K}}(1|3)$ and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals on the super-coalgebra $\widehat{\mathcal{K}}{(1|3)}^{*}$, and construct a related infinite hierarchy of completely integrable super-Hamiltonian systems on smooth functional supermanifolds, which also prove to be supersymmetric. Moreover, we generalize these results subject to the suitably factorized super-pseudo-differential Lax-type linear problems, taking into account the devised-before algebro-analytic constructions in both above-mentioned works [12,15,16,30] and in [32,33,34,35], devoted to Lie-superalgebraic properties of factorized Lax-type representations and the factorized Hamiltonian systems, respectively. As a new interesting result, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of centrally extended pseudo-differential operator superalgerbras.

2. Differential-Geometric Structures on Supercircle ${\displaystyle {\mathbb{S}}^{1|N}}$

Consider the usual one-dimensional circle ${\mathbb{S}}^1$ and its supermanifold [36,37] extension ${\mathbb{S}}^{1|N}$ by means of a coordinate variable $\left(x,\theta \right)\in {\mathbb{S}}^{1|N}\simeq {\mathbb{S}}^1\times {\mathsf{\Lambda }}^{(N)}$, specified by the ${\mathbb{Z}}_2$-graded Grassmann algebra ${\mathsf{\Lambda }}^{(N)}={\mathsf{\Lambda }}_0^{(N)}\oplus {\mathsf{\Lambda }}_1^{(N)}$ over the field $\mathbb{R}(\mathbb{C})$ with parities ${p|}_{{\mathsf{\Lambda }}_s^{(N)}}=s,$ $s\in \{0,1\}$, where $x\in {\mathbb{S}}^1,$ $\theta =({\theta }_1,{\theta }_2,...,{\theta }_N)\in {\mathsf{\Lambda }}_1^{(N)}$ and ${\theta }_j{\theta }_k+{\theta }_k{\theta }_j=0,j,k=\overline{1,N},N\in \mathbb{N}$, respectively. An arbitrary smooth uniform function $f\in C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})\simeq C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}^{(N)})\times {\mathsf{\Lambda }}^{(N)},p(f)$ $\in \{0,1\}$ is at point $\left(x,\theta \right)\in {\mathbb{S}}^1\times {\mathsf{\Lambda }}_1^{(1)}$, representable as

$$f(x,\theta )=f_0(x)+\sum _{1\le j_1<j_2<...<j_k\le N}{\theta }_{j_1}{\theta }_{j_2},...,{\theta }_{j_k}{f}_{j_1{j}_2...j_k}(x),$$

(1)

where the mappings $f_0,...,$ $f_{j_1{j}_2...j_k}\in C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_{k\mathrm{mod}2}^{(N)})$ and their parities $p(f_{j_1{j}_2...j_k})=p(f)+k\mathrm{mod}2,j\in \mathrm{Maps}(\overline{1,k};\{\overline{1,\mathrm{N}}\}),k\in \overline{1,\mathrm{N}}$. The linear space of functions (1) over the ${\mathbb{Z}}_2$-graded Grassmann algebra ${\mathsf{\Lambda }}^{(N)}$ generates the ${\mathbb{Z}}_2$-graded algebra $C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$, and the linear subspace of functions (1) with component $f_0(x)=0,x\in {\mathbb{S}}^1$ generates its nilpotent ideal $J({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})\subset C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$. It is also easy to observe that the factor space $C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})/J({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$ $\simeq C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}^{(N)})$, being equivalent to the space of coefficients of the algebra $C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$.

Consider now a diffeomorphism of ${\mathbb{S}}^{1|N}$, which is, by definition, a parity preserving the algebra automorphism of $C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$, inducing a related homomorphism of $C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}^{(N)})$. The corresponding linear space $\mathcal{G}(1,N)$ $:=Vect({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$ of vector fields on ${\mathbb{S}}^{1|N}$ is, by definition, the Lie superalgebra of all derivations of the superalgebra $C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{1,N})$; that is, for any uniform vector field $a\in \mathcal{G}(1|N)$ with parity $p(a)\in \{0,1\}$, the condition $a(fg)=a(f)g+{(-1)}^{p(a)p(f)}fa(g)$ holds for any uniform function $f\in C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$, $p(f)\in \{0,1\}$, and $g\in C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$. The Lie superalgebras, $\mathcal{G}(1|N)$-graded commutators, and any uniform elements $a,b\in \mathcal{G}(1|N)$, can be recalculated as

$$[a,b](f)=a(b(f))-{(-1)}^{p(a)p(b)}b(a(f)),$$

(2)

where $f\in C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$. The above-constructed Lie superalgebra $\mathcal{G}(1|N)$ satisfies the super-Leibnitz commutator relationships

$$\begin{array}{cc}\hfill [a,b]& =-{(-1)}^{p(a)p(b)}[b,a],\hfill \\ \hfill [a,[b,c]]& =[[a,b],c]+{(-1)}^{p(a)p(b)}[b,[a,c]]\hfill \end{array}$$

for arbitrary $a,b$, and $c\in \mathcal{G}(1|N)$, and is generated by sections $\Gamma ({\mathbb{S}}^{1|N})$ of the tangent bundle $\left(T({\mathbb{S}}^{1|N}),\pi ,{\mathbb{S}}^{1|N}\right)$ over the supercircle ${\mathbb{S}}^{1|N}$, being equivalent to the free left $C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$-module with the basis $\left(\mathsf{\partial }/\mathsf{\partial }x,\mathsf{\partial }/\mathsf{\partial }{\theta }_1,\mathsf{\partial }/\mathsf{\partial }{\theta }_2,...,\mathsf{\partial }/\mathsf{\partial }{\theta }_N\right)$ and parities $p(\mathsf{\partial }/\mathsf{\partial }x)=0,p(\mathsf{\partial }/\mathsf{\partial }{\theta }_j)=1,j=\overline{1,N}$, respectively. The adjoint space ${\mathsf{\Lambda }}^1({\mathbb{S}}^{1|N}):=\mathcal{G}{(1|N)}^{*}$ of differential 1-forms on the supercircle ${\mathbb{S}}^{1|N}$ is a free right $C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$-module with the basis $\left(dx,d{\theta }_1,d{\theta }_2,...,d{\theta }_N\right)$ and parities $p(dx)=0,p(d{\theta }_j)=1,j=\overline{1,N}$, respectively. The duality between these spaces is determined by means of the internal super-differentiation $i_a:{\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})\to {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})\simeq C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)})$, which, for any $a\in \mathcal{G}(1|N)$, is defined by the following relationships:

$$i_{\mathsf{\partial }/\mathsf{\partial }x}dx=1,i_{\mathsf{\partial }/\mathsf{\partial }x}d{\theta }_j=0,i_{\mathsf{\partial }/\mathsf{\partial }{\theta }_j}dx=0,i_{\mathsf{\partial }/\mathsf{\partial }{\theta }_j}d{\theta }_k={\delta }_{j,k}$$

(3)

for $j,k=\overline{1,N}$. We recall here that, if the $\mathsf{\Omega }({\mathbb{S}}^{1|N}):={\oplus }_{j\in {\mathbb{Z}}_{+}}{\mathsf{\Omega }}^j({\mathbb{S}}^{1|N})$ is the corresponding $\mathbb{Z}$-graded Grassmann superalgebra with ${\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})\simeq C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(N)}),{\mathsf{\Omega }}^j({\mathbb{S}}^{1|N})\simeq \underset{j\mathrm{times}}{\underbrace{{\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})\wedge {\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})\wedge ...\wedge {\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})}},j\in {\mathbb{Z}}_{+}$, where "$\wedge$" denotes the usual external multiplication on ${\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})$, the internal differentiation $i_a:{\mathsf{\Omega }}^{j+k}({\mathbb{S}}^{1|N})\to {\mathsf{\Omega }}^{j+k-1}({\mathbb{S}}^{1|N}),a\in \mathcal{G},j+k\in {\mathbb{Z}}_{+},$ acts the following way:

$$i_a\left({\alpha }^{(j)}\wedge {\alpha }^{(k)}\right)=(i_a{\alpha }^{(j)})\wedge {\alpha }^{(k)}+{(-1)}^{j+p(a)p({\alpha }^{(j)})}{\alpha }^{(j)}\wedge (i_a{\alpha }^{(k)}),i_a^2=0,$$

(4)

for arbitrary differential forms ${\alpha }^{(j)}\in {\mathsf{\Omega }}^j({\mathbb{S}}^{1|N})$ and ${\alpha }^{(k)}\in {\mathsf{\Omega }}^k({\mathbb{S}}^{1|N}),j,k\in {\mathbb{Z}}_{+}$, and is an anti-differentiation [38] of the $(-1)$-degree. Respectively, the external differentiation $d:$ ${\mathsf{\Omega }}^{j+k}({\mathbb{S}}^{1|N})\to$ ${\mathsf{\Omega }}^{j+k+1}({\mathbb{S}}^{1|N}),j+k\in \mathbb{N},$ acts as

$$d\left({\alpha }^{(j)}\wedge {\alpha }^{(k)}\right)=(d{\alpha }^{(j)})\wedge {\alpha }^{(k)}+{(-1)}^j{\alpha }^{(j)}\wedge (d{\alpha }^{(k)}),d^2=0,$$

(5)

where, in particular, ${\alpha }^{(j)}\wedge {\alpha }^{(k)}={(-1)}^{jk+p(a)p(b)}{\alpha }^{(k)}\wedge {\alpha }^{(j)}$ for arbitrary differential forms ${\alpha }^{(j)}\in {\mathsf{\Omega }}^j({\mathbb{S}}^{1|N})$ and ${\alpha }^{(k)}\in {\mathsf{\Omega }}^k({\mathbb{S}}^{1|N})$, and is an anti-differentiation [38] of the $(+1)$-degree. A combination of these two anti-differentiations, owing to the Cartan identity

$$i_{a}d+d{i}_a={\mathcal{L}}_a,$$

(6)

coincides [38] for any vector field $a\in \mathcal{G}(1|N)$ with the Lie derivative of the Grassmann algebra $\mathsf{\Omega }({\mathbb{S}}^{1|N})$.

Introduce now the so-called canonical super-derivations $D_j:=D_{{\theta }_j}=\mathsf{\partial }/\mathsf{\partial }{\theta }_j+{\theta }_j\mathsf{\partial }/\mathsf{\partial }x\in \mathcal{G}(1|N),$ $j=\overline{1,N},$ satisfying the following relationships:

$$[D_j,D_k]=D_j{D}_k+D_k{D}_j=2{\delta }_{j,k},D_j^2=\mathsf{\partial }/\mathsf{\partial }x$$

(7)

for all $j,k=\overline{1,N}$, and describe all vector fields $K_f\in \mathcal{G}(1|N),f\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$, leaving invariant the following contact differential one-form ${\alpha }^{(1)}\in {\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})$:

$${\alpha }^{(1)}:=dx+\sum _{j=\overline{1,N}}{\theta }_{j}d{\theta }_j,$$

(8)

that is,

$${\mathcal{L}}_{K_f}{\alpha }^{(1)}={\mu }_f{\alpha }^{(1)}$$

(9)

for some mapping ${\mu }_f\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$. Taking into account the differential-geometric relationships

$$i_{D_j}{\mathcal{L}}_{K_f}{\alpha }^{(1)}=i_{[D_j,K_f]}{\alpha }^{(1)},i_{D_j}{\alpha }^{(1)}=0,$$

(10)

for any $j=\overline{1,N}$, one easily obtains that

$$K_f=f\mathsf{\partial }/\mathsf{\partial }x+1/2{(-1)}^{p(f)}\langle Df|D\rangle ,{\mu }_f=-1/2\mathsf{\partial }f/\mathsf{\partial }x,$$

(11)

for any smooth uniform mapping $f\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$, where $D=(D_1,D_2,...,D_N)$ is the so-called super-gradient on ${\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$ and $\langle ·|·\rangle$ is the usual bilinear form, mimicking that on ${\mathbb{C}}^N\otimes {\mathbb{C}}^N$. As a natural consequence of the invariance (9), one derives that the set $\mathcal{K}(1|N):=\{K_f\in \mathcal{G}(1|N):f\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})\}$-the Lie superalgebra of supervector fields on ${\mathbb{S}}^{1|N}$, called the conformal superalgebra of ${\mathbb{S}}^{1|N}$—contactomorphisms. Namely, the group $\mathrm{Cont}({\mathbb{S}}^{1|N})$ of the corresponding contactomorphisms $\mathsf{\Phi }:{\mathbb{S}}^{1|N}\to {\mathbb{S}}^{1|N}$ satisfies the condition ${\mathsf{\Phi }}^{*}{\alpha }^{(1)}={\eta }_{\mathsf{\Phi }}{\alpha }^{(1)}$ for some mapping ${\eta }_{\mathsf{\Phi }}\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$. In particular, the following functional relationship $({\mu }_f\circ \mathsf{\Phi }-{\mu }_f){\eta }_{\mathsf{\Phi }}=K_f({\eta }_{\mathsf{\Phi }})$ for arbitrary mappings $\mathsf{\Phi }\in \mathrm{Cont}({\mathbb{S}}^{1|N})$ and $f\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$ holds. Moreover, for any uniform functions $f,g\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$, the following Lie super-commutator expression holds:

$$[K_f,K_g]=K_{\{f,g\}},$$

(12)

where

$$\{f,g\}=f\mathsf{\partial }g/\mathsf{\partial }x-{(-1)}^{p(f)p(g)}g\mathsf{\partial }f/\mathsf{\partial }x+1/2{(-1)}^{p(f)p(g)}\langle Df|Dg\rangle ,$$

(13)

meaning that the mapping

$$K:({\mathsf{\Omega }}^0({\mathbb{S}}^{1|N}),\{·,·\})\to (\mathcal{K}(1|N),[·,·])$$

(14)

is a Lie superalgebra isomorphism. The latter makes it possible to identify these Lie superalgebras—$({\mathsf{\Omega }}^0({\mathbb{S}}^{1|N}),\{·,·\})\simeq (\mathcal{K}(1|N),[·,·])$—that will be exploited in what follows below. It is also useful to remark here that the superconformal Lie superalgebra $\mathcal{K}(1|N)\subset \mathcal{G}(1|N)$ gives rise to the inverse imbedding $\mathcal{G}{(1|N)}^{*}\subset \mathcal{K}{(1|N)}^{*}$; thus, the problem of representation of the super-coalgebra $\mathcal{K}{(1|N)}^{*}$ arises as nontrivial enough. Moreover, if one assumes that there exists some bilinear form ${(·|·)}_c:\mathcal{K}{(1|N)}^{*}\times \mathcal{K}(1|N)\to \mathbb{C}$, it can be isometrically related to the canonical bilinear form $(·|·):{\mathsf{\Omega }}^0{({\mathbb{S}}^{1|N})}^{*}\times {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})\to \mathbb{C}$ in the following natural way:

$${(r(l)|K_f)}_c=(l|f):={\int }_0^{2\pi }dx\int d\theta f(x,\theta )l(x,\theta )$$

(15)

for arbitrary $f\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N}),p(f)=0$ and $l\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N}),p(l)=N\mathrm{mod}(2)$. Here, the super-integration $\int dx\int d\theta (·)$ “measure” is defined [36] for all $j,k=\overline{1,N}$ via the following rules:

$$\int {\theta }_{j}d{\theta }_k={\delta }_{j,k},\int d{\theta }_j=0,$$

(16)

and the linear mapping $r:$ ${\mathsf{\Omega }}^0{({\mathbb{S}}^{1|N})}^{*}\to$ $\mathcal{K}{(1|N)}^{*}\simeq {\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})$ is determined from (15) and the Lie superalgebra isomorphism (14) as follows:

$$r(l)={\left(K^{*}\right)}^{-1}l$$

(17)

for arbitrary element $l\in {\mathsf{\Omega }}^0{({\mathbb{S}}^{1|N})}^{*}$. Taking into account that the right-hand side of relationship (15) is invariant with respect to the group of contactomorphisms $\mathrm{Cont}({\mathbb{S}}^{1|N})$, it is enough to check that the left-hand side of (15) is invariant too; that is, ${\mathcal{L}}_{K_g}{(r(l)|K_f)}_s=0$ for any $f,g\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$. The latter immediately reduces to the following condition: ${\mathcal{L}}_{K_g}(r^{*}(l))=a{d}_{K_g}^{*}r(l)$ for arbitrary $g\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$ and $l\in {\mathsf{\Omega }}^0{({\mathbb{S}}^{1|N})}^{*}$, which basically allows us to construct the generalized differential one-form mapping $r:{\mathsf{\Omega }}^0{({\mathbb{S}}^{1|N})}^{*}\to \mathcal{K}{(1|N)}^{*}\simeq {\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})$, albeit on that we will not dwell in detail.

3. Casimir Invariants, Coadjoint Orbits of the Lie Superalgebra ${\displaystyle \mathcal{K}(1|3)}$, and Related Integrable Hamiltonian Supersystems

Consider now the conformal Lie superalgebra $\mathcal{K}(1|N)\simeq {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})$ for $N\in \mathbb{N}$ and its affine extension $\tilde{\mathcal{K}}(1|N):=\mathcal{K}(1|N)\otimes \{\lambda ,{\lambda }^{-1}),\lambda \in \mathbb{C}$, allowing for the direct Lie super-subalgebra splitting $\tilde{\mathcal{K}}(1|N)={\tilde{\mathcal{K}}}_{+}(1|N)\oplus {\tilde{\mathcal{K}}}_{-}(1|N)$, where, by definition,

$$\begin{array}{cc}\hfill {\tilde{\mathcal{K}}}_{+}(1|N)& \simeq \{\tilde{a}(x,\theta ;\lambda )=\sum _{j=0}^{n\ll \infty }{a}_j(x,\theta ){\lambda }^j:(x,\theta )\in {\mathbb{S}}^{1|N},a_j\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})\simeq \mathcal{K}(1|N),j=\overline{0,n}\},\hfill \\ \hfill {\tilde{\mathcal{K}}}_{-}(1|N)& \simeq \{\tilde{b}(x,\theta ;\lambda )=\sum _{j\in \mathbb{N}}{b}_j(x,\theta ){\lambda }^{-j}:(x,\theta )\in {\mathbb{S}}^{1|N},b_j\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})\simeq \mathcal{K}(1|N),j\in \mathbb{N}\},\hfill \end{array}$$

(18)

owing to the isomorphism (14). The affine super-coalgebra $\tilde{\mathcal{K}}{(1|N)}^{*}\simeq {\mathsf{\Omega }}^1({\mathbb{S}}^{1|N})\otimes \{\lambda ,{\lambda }^{-1}\}\simeq {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})\otimes \{\lambda ,{\lambda }^{-1}\}$ naturally related to $\tilde{\mathcal{K}}(1|N)$ is defined via the following nondegenerate bilinear form:

$$(\tilde{l}|\tilde{a})=re{s}_{\lambda \in \mathbb{C}}{\int }_0^{2\pi }dx\int d\theta \tilde{l}(x,\theta ;\lambda )\tilde{a}(x,\theta ;\lambda )$$

(19)

for arbitrary elements $\tilde{a}={\sum }_{j\ll \infty }{a}_j(x,\theta ){\lambda }^j\in \tilde{\mathcal{K}}(1|N)$ and $\tilde{l}={\sum }_{j\ll \infty }{l}_j(x,\theta ){\lambda }^j\in \tilde{\mathcal{K}}{(1|N)}^{*}$, where the super-integration $\int d\theta (·)$ is performed [36] using the rules (16) for $j,k=\overline{1,N}$.

It is well known [39,40] that the only superconformal Lie superalgebras $\mathcal{K}(1|N)$ for $N=1,2$, and 3 possess the central extensions $\mathcal{K}(1|N)\oplus {\mathsf{\Lambda }}_0^{(N)},$ $N=\overline{1,3}$ by means of the following cocycles:

$$\begin{array}{cc}\hfill {\omega }_1^{(1)}(a,b)& ={\int }_0^{2\pi }dx\int d\theta a(x,\theta )D_1^{5}b(x,\theta ),\hfill \\ \hfill {\omega }_1^{(2)}(a,b)& ={\int }_0^{2\pi }dx\int d\theta a(x,\theta )D_1{D}_2\mathsf{\partial }b(x,\theta )/\mathsf{\partial }x,\hfill \\ \hfill {\omega }_1^{(3)}(a,b)& ={\int }_0^{2\pi }dx\int d\theta a(x,\theta )D_1{D}_2{D}_{3}b(x,\theta ),\hfill \end{array}$$

(20)

for any $a,b\in {\mathsf{\Omega }}^0({\mathbb{S}}^{1|N})\simeq \mathcal{K}(1|N),N=\overline{1,3},$ respectively, satisfying the determining relationships

$$\begin{array}{cc}\hfill {\omega }_1^{(N)}(a,b)& =-{(-1)}^{p(a)p(b)}{\omega }_1^{(N)}(b,a),\hfill \\ \hfill {\omega }_1^{(N)}(a,[b,c])& ={\omega }_1^{(N)}([a,b],c)+{(-1)}^{p(a)p(b)}{\omega }_1^{(N)}(b,[a,c])\hfill \end{array}$$

(21)

for arbitrary uniform vector fields $a,b$, and $c\in \mathcal{K}(1|N),$ $N=\overline{1,3}$.

Thus, the centrally extended Lie affine superalgebras $\widehat{\mathcal{K}}(1|N):=\tilde{\mathcal{K}}(1|N)\oplus {\mathsf{\Lambda }}_0^{(N)},$ $N=\overline{1,3},$ respectively, are defined by means of the related-to-(20) cocycles

$$\begin{array}{cc}\hfill {\omega }_1^{(1)}(\tilde{a},\tilde{b})& ={\mathrm{res}}_{\lambda \in \mathbb{C}}{\int }_0^{2\pi }dx\int d\theta \tilde{a}(x,\theta )D_1^5\tilde{b}(x,\theta ),\hfill \\ \hfill {\omega }_1^{(2)}(\tilde{a},\tilde{b})& ={\mathrm{res}}_{\lambda \in \mathbb{C}}{\int }_0^{2\pi }dx\int d\theta \tilde{a}(x,\theta )D_1{D}_2\mathsf{\partial }\tilde{b}(x,\theta )/\mathsf{\partial }x,\hfill \\ \hfill {\omega }_1^{(3)}(\tilde{a},\tilde{b})& ={\mathrm{res}}_{\lambda \in \mathbb{C}}{\int }_0^{2\pi }dx\int d\theta \tilde{a}(x,\theta )D_1{D}_2{D}_3\tilde{b}(x,\theta ),\hfill \end{array}$$

(22)

subject to which the corresponding Lie super-commutators equal

$$[(\tilde{a},\alpha ),(\tilde{b},\beta )]:=([\tilde{a},\tilde{b}],{\omega }_1^{(N)}(\tilde{a},\tilde{b}))$$

(23)

for arbitrary $(\tilde{a},\alpha ),(\tilde{b},\beta )\in \widehat{\mathcal{K}}(1|N),N=\overline{1,3}$. Now, let $\widehat{\mathcal{K}}{(1|N)}^{*}\simeq \tilde{\mathcal{K}}{(1|N)}^{*}\oplus {\mathsf{\Lambda }}_0^{(N)},N=\overline{1,3},$ denote the adjoint to the Lie superalgebras $\widehat{\mathcal{K}}(1|N),N=\overline{1,3},$ and super-coalgebras, respectively, defined by means of the following nondegenerate bilinear form:

$$((\tilde{l},k)|(\tilde{a},\alpha ))=(\tilde{l}|\tilde{a})+k\alpha$$

(24)

for arbitrary $(\tilde{l},k)\in \widehat{\mathcal{K}}{(1|N)}^{*}$ and $(\tilde{a},\alpha )\in \widehat{\mathcal{K}}(1|N),N=\overline{1,3}$. Subject to the bilinear form (24), one can determine the corresponding coadjoint actions $a{d}^{*}:\widehat{\mathcal{K}}(1|N)\times \widehat{\mathcal{K}}{(1|N)}^{*}\to \widehat{\mathcal{K}}{(1|N)}^{*}$ of the Lie superalgebras $\tilde{\mathcal{K}}(1|N),N=\overline{1,3},$ on the adjoint spaces $\widehat{\mathcal{K}}{(1|3)}^{*},N=\overline{1,3},$ by means of the following relationships:

$$(-a{d}_{(\tilde{a},\alpha )}^{*}(\tilde{l},k)|(\tilde{b},\beta ))=(\tilde{l}|[\tilde{a},\tilde{b}])+k{\omega }_1^{(N)}(\tilde{a},\tilde{b})$$

(25)

for fixed $(\tilde{a},\alpha )\in \widehat{\mathcal{K}}(1|N),(\tilde{l},k)\in \widehat{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3},$ and arbitrary $(\tilde{b},\beta )\in \widehat{\mathcal{K}}(1|N),N=\overline{1,3}$. As a result of simple calculations, we obtain from (25) that

$$a{d}_{(\tilde{a},\alpha )}^{*}(\tilde{l},k)=(a{d}_{\tilde{a}}^{*}\tilde{l}+k{D}_1^5\tilde{a},0),$$

(26)

for any $(\tilde{a},\alpha )\in \widehat{\mathcal{K}}(1|1)$ and $(\tilde{l},k)\in \widehat{\mathcal{K}}{(1|1)}^{*}$,

$$a{d}_{(\tilde{a},\alpha )}^{*}(\tilde{l},k)=(a{d}_{\tilde{a}}^{*}\tilde{l}+k{D}_1{D}_2\mathsf{\partial }\tilde{a}/\mathsf{\partial }x,0),$$

(27)

for any $(\tilde{a},\alpha )\in \widehat{\mathcal{K}}(1|2)$ and $(\tilde{l},k)\in \widehat{\mathcal{K}}{(1|2)}^{*}$ and

$$a{d}_{(\tilde{a},\alpha )}^{*}(\tilde{l},k)=(a{d}_{\tilde{a}}^{*}\tilde{l}+k{D}_1{D}_2{D}_3\tilde{a},0),$$

(28)

for any $(\tilde{a},\alpha )\in \widehat{\mathcal{K}}(1|3)$ and $(\tilde{l},k)\in \widehat{\mathcal{K}}{(1|3)}^{*}$.

Being interested in describing commuting hierarchies of evolution superflows on $\widehat{\mathcal{K}}{(1|N)}^{*}\simeq \tilde{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3},$ within the classical Adler–Kostant–Souriau Lie-algebraic scheme [41,42,43,44,45,46,47], we need to construct the sets $I(\tilde{\mathcal{K}}{(1|N)}^{*}),N=\overline{1,3},$ of smooth Casimir functionals $\gamma :\tilde{\mathcal{K}}{(1|N)}^{*}\to {\mathsf{\Lambda }}_0^{(N)},N=\overline{1,3},$ respectively, invariant with respect to the coadjoint actions (26)–(28):

$$a{d}_{\mathrm{grad}\gamma (\tilde{l})}^{*}(\tilde{l},k)=0\sim a{d}_{\mathrm{grad}\gamma (\tilde{l})}^{*}\tilde{l}+1/2D_1^5\mathrm{grad}\gamma (\tilde{l})=0$$

(29)

for $N=1$,

$$a{d}_{\mathrm{grad}\gamma (\tilde{l})}^{*}(\tilde{l},k)=0\sim a{d}_{\mathrm{grad}\gamma (\tilde{l})}^{*}\tilde{l}+1/2D_1{D}_2\mathsf{\partial }\mathrm{grad}\gamma (\tilde{l})/\mathsf{\partial }x=0$$

(30)

for $N=2$, and

$$a{d}_{\mathrm{grad}\gamma (\tilde{l})}^{*}(\tilde{l},k)=0\sim a{d}_{\mathrm{grad}\gamma (\tilde{l})}^{*}\tilde{l}+1/2D_1{D}_2{D}_3\mathrm{grad}\gamma (\tilde{l})=0$$

(31)

for $N=3$, where we put, for brevity, $k=1/2$, and define, respectively, the gradient elements $\phi (\tilde{l}):=\mathrm{grad}\gamma (\tilde{\mathrm{l}})\in \tilde{\mathcal{K}}(1|N),N=\overline{1,3},$ $p(\phi (\tilde{l}))=0$ at fixed uniform points $\tilde{l}\in \tilde{\mathcal{K}}{(1|N)}^{*},p(\tilde{l})=N\mathrm{mod}2,N=\overline{1,3},$ via the following common relationship:

$$(\tilde{m}|\mathrm{grad}\gamma (\tilde{\mathrm{l}})):=d\gamma (\tilde{\mathrm{l}}+\epsilon \tilde{\mathrm{m}}){/\mathrm{d}\epsilon |}_{\epsilon =0},$$

(32)

satisfied for all $\tilde{m}\in \tilde{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3}$. Calculating also the coadjoint actions $a{d}_{\mathrm{grad}\gamma (\tilde{\mathrm{l}})}^{*}\tilde{l}\in \tilde{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3},$ at the chosen uniform point $\tilde{l}\in \tilde{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3},$ as

$$a{d}_{\mathrm{grad}\gamma (\tilde{\mathrm{l}})}^{*}\tilde{l}={\displaystyle \frac{\mathsf{\partial }\tilde{l}}{\mathsf{\partial }x}}x\mathrm{grad}\gamma (\tilde{l})+2(1-N/4)\tilde{l}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\mathrm{grad}\gamma (\tilde{l})+1/2{(-1)}^{N+1}\langle D\tilde{l}|D\mathrm{grad}\gamma (\tilde{l})\rangle$$

(33)

for one ensures from (29)–(31) and (33) that the gradient elements $\phi (\tilde{l}):=\mathrm{grad}\gamma (\tilde{l})\in \tilde{\mathcal{K}}(1|N),N=\overline{1,3},$ satisfy [13] the following determining differential-functional equations:

$$1/2D_1^5\phi (\tilde{l})+\mathsf{\partial }\tilde{l}/\mathsf{\partial }x\phi (\tilde{l}))+3/2\tilde{l}\mathsf{\partial }\phi (\tilde{l})/\mathsf{\partial }x+1/2\langle D\tilde{l}|D\phi (\tilde{l})\rangle =0$$

(34)

for $N=1,p(\tilde{l})=1$,

$$1/2D_1{D}_2\mathsf{\partial }\phi (\tilde{l})/\mathsf{\partial }x+\mathsf{\partial }\left(\tilde{l}/\mathsf{\partial }x\phi (\tilde{l})\right)/\mathsf{\partial }x-1/2\langle D\tilde{l}|D\phi (\tilde{l})\rangle =0$$

(35)

for $N=2,p(\tilde{l})=0$, and

$$1/2D_1{D}_2{D}_3\phi (\tilde{l})+\mathsf{\partial }\tilde{l}/\mathsf{\partial }x\phi (\tilde{l})+1/2\tilde{l}\mathsf{\partial }\phi (\tilde{l})/\mathsf{\partial }x+1/2\langle D\tilde{l}|D\phi (\tilde{l})\rangle =0$$

(36)

for $N=3,p(\tilde{l})=1$, respectively. Now take into account that, owing to the Lie super-subalgebra splitting (18), the centrally extended affine Lie superalgebras $\widehat{\mathcal{K}}(1|N),N=\overline{1,3},$ possess another Lie superalgebra commutator,

$${[(\tilde{a},\alpha ),(\tilde{b},\beta )]}_R:=({[\tilde{a},\tilde{b}]}_R,{\omega }_1(R\tilde{a},\tilde{b})+{\omega }_1^{(N)}(\tilde{a},R\tilde{b})),$$

(37)

modified [42,44,48,49] by means of the R-structure homomorphism $R:=1/2(P_{+}-P_{-}):\tilde{\mathcal{K}}(1|N)\to \tilde{\mathcal{K}}(1|N),N=\overline{1,3},$ respectively, where, by definition, ${[\tilde{a},\tilde{b}]}_R:=[R\tilde{a},\tilde{b}]+[\tilde{a},R\tilde{b}]$ and $P_{\pm }$ $\tilde{\mathcal{K}}(1|N):=$ ${\tilde{\mathcal{K}}}_{\pm }(1|N)\subset \tilde{\mathcal{K}}(1|N),N=\overline{1,3},$ are the corresponding projectors on the Lie superalgebras $\tilde{\mathcal{K}}(1|N),N=\overline{1,3}$. The Lie superalgebra structures (23) and (25) generate, respectively, the following compatible [50] Lie-=Poisson structures [42,44,48,49,51]:

$${\{\gamma ,\mu \}}_{Lie-P}:=(\tilde{l}|[\mathrm{grad}\gamma (\tilde{\mathrm{l}}),\mathrm{grad}\mu (\tilde{\mathrm{l}})])+{\omega }_1^{(N)}(\mathrm{grad}\gamma (\tilde{\mathrm{l}}),\mathrm{grad}\mu (\tilde{\mathrm{l}}))$$

(38)

and

$$\begin{array}{c}\{\gamma ,\mu \}:=(\tilde{l}|{[\mathrm{grad}\gamma (\tilde{\mathrm{l}}),\mathrm{grad}\mu (\tilde{l})]}_R)+\\ \\ +{\omega }_1^{(N)}(R\mathrm{grad}\gamma (\tilde{\mathrm{l}}),\mathrm{grad}\mu (\tilde{\mathrm{l}}))+{\omega }_1^{(N)}(\mathrm{grad}\gamma (\tilde{\mathrm{l}}),R\mathrm{grad}\mu (\tilde{\mathrm{l}}))\end{array}$$

(39)

on the coadjoint superspaces $\widehat{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3}$, where, by definition, $\tilde{l}:={\sum }_{s\ll \infty }{l}_s(x;\theta ){\lambda }^{-s}\in \tilde{\mathcal{K}}{(1|3)}^{*},$ $\mathrm{grad}{\gamma }_j(\tilde{l})={\lambda }^j\phi (\tilde{l})\in \tilde{\mathcal{K}}(1|N),N=\overline{1,3},j\in \mathbb{N}$, and $\phi (\tilde{l})\sim {\sum }_{k\in {\mathbb{Z}}_{+}}{\phi }_k(l){\lambda }^{-k}$ $\in \tilde{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3},$ are the special solutions to the determining differential-functional Equations (34)–(36). Let us now construct a functional Hilbert-type superspace $L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0^{(N)})$ with the norm

$${||f||}_{L_2}:={\left({\int }_0^{2\pi }dx\int d\theta {|f(x,\theta ;\lambda )|}^2\right)}^{1/2}\ne \infty$$

(40)

for functions $f\in L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0^{(N)}),N=\overline{1,3},$ $p(f)=0$. Then, the above-constructed countable sets $I(\tilde{\mathcal{K}}{(1,N)}^{*}),N=\overline{1,3},$ of Casimir functionals are characterized within the Adler–Kostant–Symes scheme [41,43,47] by the following proposition.

Proposition 1. 

The Casimir invariants $I(\tilde{\mathcal{K}}{(1|N)}^{*}),N=\overline{1,3},$ compile as involutive with respect to the Lie–Poisson bracket (39) sets of functionals, generating on the coadjoint spaces $\widehat{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3},$ counted hierarchies

$$\mathsf{\partial }\tilde{l}/\mathsf{\partial }{t}_j=-1/2D_1^5{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}-\mathsf{\partial }\tilde{l}/\mathsf{\partial }x{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}-3/2\tilde{l}\mathsf{\partial }{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}/\mathsf{\partial }x-1/2\langle D\tilde{l}|D{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}\rangle$$

(41)

for $N=1,j\in \mathbb{N}$,

$$\mathsf{\partial }\tilde{l}/\mathsf{\partial }{t}_j=-1/2D_1{D}_2\mathsf{\partial }{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}/\mathsf{\partial }x-\mathsf{\partial }\left(\tilde{l}/\mathsf{\partial }x{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}\right)/\mathsf{\partial }x+1/2\langle D\tilde{l}|D{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}\rangle$$

(42)

for $N=2,j\in \mathbb{N}$, and

$$\mathsf{\partial }\tilde{l}/\mathsf{\partial }{t}_j=-1/2D_1{D}_2{D}_3{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}-\mathsf{\partial }\tilde{l}/\mathsf{\partial }x{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}-1/2\tilde{l}\mathsf{\partial }{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}/\mathsf{\partial }x-1/2\langle D\tilde{l}|D{\left({\lambda }^j\phi (\tilde{l})\right)}_{+}\rangle$$

(43)

for $N=3,j\in \mathbb{N}$, respectively, commuting to each other Liouville integrable Hamiltonian superflows. Moreover, as it simply follows from the determining differential-functional Equations (34)–(36), all the flows (41)–(43) are generated, respectively, by the following linear integro-differential super-Sturm–Liouville-type [13,14] spectral problems

$$L^{(1)}(x,\theta ;D,\lambda )f(x,\theta ;\lambda ):=D_1^{3}f(x,\theta ;\lambda )/\mathsf{\partial }x+{\tilde{l}}_m(w;\lambda )(x,\theta )f(x,\theta ;\lambda )=0$$

(44)

for $N=1$ on the superspace $L_2({\mathbb{S}}^{1|1};{\mathsf{\Lambda }}^{(1|1)})$,

$$L^{(2)}(x,\theta ;D,\lambda )f(x,\theta ;\lambda ):=D_1{D}_{2}f(x,\theta ;\lambda )+{\tilde{l}}_m(w;\lambda )(x,\theta )f(x,\theta ;\lambda )=0$$

(45)

for $N=2$ on the superspace $L_2({\mathbb{S}}^{1|2};{\mathsf{\Lambda }}^{(1|2)})$, and

$$L^{(3)}(x,\theta ;D,\lambda )f(x,\theta ;\lambda ):=D_1{D}_2{D}_3\mathsf{\partial }/\mathsf{\partial }{x}^{-1}f(x,\theta ;\lambda )+{\tilde{l}}_m(w;\lambda )(x,\theta )f(x,\theta ;\lambda )=0$$

(46)

for $N=3$ on the superspace $L_2({\mathbb{S}}^{1|3};{\mathsf{\Lambda }}^{(1|3)})$, specified by the super-potential

$${\tilde{l}}_p(w;\lambda )(x,\theta )=\sum _{j\in \mathbb{N}}{w}_j(x,\theta ){\lambda }^{p-j}$$

(47)

at points $(x,\theta )\in {\mathbb{S}}^{1|N}$, where $p\in \mathbb{N}$ is an arbitrary yet fixed integer and $w_j:{\mathbb{S}}^{1|N}\to {\mathsf{\Lambda }}_{N\mathit{mod}(2)}^{(N)},$ $N=\overline{1,3},j\in \mathbb{N},$ are chosen smooth mappings.

The generalized [52,53,54] spectrum

$$\sigma ({\tilde{l}}_p):={\{\lambda \in \mathbb{C}:||f||}_2\ne \infty \}$$

(48)

of the super-Sturm–Liouville-type spectral problems (44)–(46) is characterized, owing to the reasonings from [55], by the following proposition.

Proposition 2. 

The generalized spectra (48) of the super-Sturm–Liouville-type spectral problems (44)–(46) have, in general, an infinite-zoned structure and are invariant with respect to the countable hierarchies of super-Hamiltonian flows (41)–(43) commuting to each other, respectively.

The generalized super-Sturm–Liouville-type spectral problems (44)–(46) are equivalent to the corresponding Lax-type linear spectral problem on $L_2({\mathbb{S}}^{1|1};{\mathsf{\Lambda }}^{(1|N)}),N=\overline{1,3},$ subject to which the countable hierarchies of superflows (41)–(43) commuting to each other reduce to the countable hierarchy of Zakharov–Shabat-type integro-differential [42,48,49,55] operator relationships

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{t}_j}}{L}^{(N)}(x,\theta ;D,\lambda )=[L^{(N)}(x,\theta ;D,\lambda ),M_j^{(N)}(x,\theta ;D,\lambda )],$$

(49)

calculated at points $(x,\theta )\in {\mathbb{S}}^{1|N},$ $N=\overline{1,3},$ where the related integro-differential operator expressions $M_j^{(N)}:L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(1|N)})$ $\to L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(1)})$ can be suitably determined by the Casimir gradients $\mathrm{grad}{\gamma }_j(\tilde{l})\in \widehat{\mathcal{K}}(1|N),j\in \mathbb{N},$ $N=\overline{1,3}$. Namely, let $S^{(N)}(x,\theta ;D,\lambda )\in \mathrm{End}({\mathsf{\Lambda }}_0^{(1|N)}\times {\mathsf{\Lambda }}_1^{(1|N)}\times {\mathsf{\Lambda }}_1^{(1|N)}))$ denote the monodromy matrix [42,49,55,56] at point $(x,\theta )\in {\mathbb{S}}^{1|N}$ of the super-differential operator $L^{(N)}(x,\theta ;D,\lambda ):L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(1|N)})\to L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(1|N)})$, satisfying the determining equation

$$\mathsf{\partial }{S}^{(N)}/\mathsf{\partial }x=[{\mathcal{L}}^{(N)},S^{(N)}],$$

(50)

where ${\mathcal{L}}^{(N)}\in \mathrm{End}({\mathsf{\Lambda }}_0^{(1|N)}\times {\mathsf{\Lambda }}_1^{(1|N)}\times {\mathsf{\Lambda }}_1^{(1|N)}))$ is the matrix representation.

Consider the simplest example, when the superpotential (47) is linear with respect to the spectral parameter $\lambda \in \mathbb{C}$; that is,

$${\tilde{l}}_1(w;\lambda )=w_0\lambda +w_1,$$

(51)

where $w_j\in C^{\infty }({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}^{(1|N)}),j\in \overline{0,1},$ $N=\overline{1,3},$ are some chosen smooth mappings.

Example 1. 

The super-KdV hierarchy and its generalizations.

Case $N=1$: Let us specify the seed element (51) by putting $w_0=-{\theta }_1,w_1=v+{\theta }_{1}u$, where $u\in C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_0^{(1)})$ and $v\in C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_1^{(1)})$. The corresponding Casimir super-gradients $\mathrm{grad}{\gamma }_j({\tilde{l}}_1),j\in {\mathbb{Z}}_{+},$ recurrently follow from (34) and equal

$$\begin{array}{cc}\hfill \mathrm{grad}{\gamma }_0[u,v]& =1,\mathrm{grad}{\gamma }_1[u,v]=1/2u+2\theta v_x,\hfill \\ \hfill \mathrm{grad}{\gamma }_2[u,v]& =-1/8u_{xx}+3/8u^2-3/2v{v}_x+2\theta (v_{3x}-3/2u{u}_x-3/4u_{x}v),...\hfill \end{array}$$

(52)

and so on. The corresponding invariants equal

$${\gamma }_0={\int }_0^{2\pi }udx,{\gamma }_1={\displaystyle \frac{1}{4}}{\int }_0^{2\pi }(u^2-v{v}_x)dx,{\gamma }_2={\displaystyle \frac{1}{16}}{\int }_0^{2\pi }(u_x^2+2u^3-16v_x{v}_{xx}-24uv{v}_x)dx,....$$

(53)

Taking into account the Poisson bracket definition (39), one easily ensues [14] from the gradient-type relationship (34) the following ultra-local [49] pair ${\{·,·\}}_{\vartheta }$ and ${\{·,·\}}_{\eta }$ of compatible super-Poisson brackets:

$$\begin{array}{cc}\hfill {\{u(x),u(y)\}}_{\vartheta }& =\mathsf{\partial }\delta (x-y)/\mathsf{\partial }x,{\{u(x),v(y)\}}_{\vartheta }=0,\hfill \\ \hfill {\{v(x),v(y)\}}_{\vartheta }& =1/4\delta (x-y)\hfill \end{array}$$

(54)

and

$$\begin{array}{cc}\hfill {\{u(x),u(y)\}}_{\eta }& =\left(-{\mathsf{\partial }}^3/2+u(x)\mathsf{\partial }+\mathsf{\partial }u(x)\right)\delta (x-y),\hfill \\ \hfill {\{u(x),v(y)\}}_{\eta }& =\left(3/2v(x)\mathsf{\partial }+1/2u_x(x)\right)\delta (x-y),{\{v(x),v(y)\}}_{\eta }=1/2\left(u-{\mathsf{\partial }}^2\right)\delta (x-y)\hfill \end{array}$$

(55)

for local functionals $(u,v)\in M^{1,1}\subset C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_0^{(1)}\times {\mathsf{\Lambda }}_1^{(1)})\simeq {\widehat{\mathcal{K}}}^{*}(1|1)$ at points $x,y\in {\mathbb{S}}^1$, where we put, for brevity, $\mathsf{\partial }:=\mathsf{\partial }/\mathsf{\partial }x,x\in {\mathbb{S}}^1$. The corresponding countable hierarchy of local super-KdV superflows on a functional supermanifold $M^{1,1}\subset C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}^{(1|1)})$ is obtained as

$$\mathsf{\partial }u/\mathsf{\partial }{t}_j:={\{{\gamma }_j,u\}}_{\eta },\mathsf{\partial }v/\mathsf{\partial }{t}_j:={\{{\gamma }_j,v\}}_{\eta },$$

(56)

which can be equivalently rewritten as

$$\mathsf{\partial }{(u,v)}^{⊺}/\mathsf{\partial }{t}_j=-\eta \mathrm{grad}{\gamma }_j[u,v]=-\vartheta \mathrm{grad}{\gamma }_{j+1}[u,v]$$

(57)

for $j\in \mathbb{N}$, where the super-Poisson operators $\vartheta ,\eta :T^{*}(M^{1,1})\to T(M^{1,1})$ are given by the next matrix expressions:

$$\vartheta =\left(\begin{array}{cc}\mathsf{\partial }& 0\\ 0& 1/4\end{array}\right),\eta =\left(\begin{array}{cc}-{\mathsf{\partial }}^3/2+u\mathsf{\partial }+\mathsf{\partial }u& 3/2v\mathsf{\partial }+1/2u_x\\ 3/2\mathsf{\partial }v-1/2u_x& 1/2\left(u-{\mathsf{\partial }}^2\right)\end{array}\right).$$

(58)

In particular, for $j=1$, one easily obtains [5,14] the super-KdV dynamical system

$$\begin{array}{cc}\hfill \mathsf{\partial }u/\mathsf{\partial }t& ={\{{\gamma }_1,u\}}_{\eta }=-u_{xxx}+6u{u}_x+12{(v{v}_x)}_x,\hfill \\ \hfill \mathsf{\partial }v/\mathsf{\partial }t& ={\{{\gamma }_1,v\}}_{\eta }=-4v_{xxx}+6u{v}_x+3u_{x}v,\hfill \end{array}$$

(59)

on the functional supermanifold $M^{1,1}$, which nonetheless is not supersymmetric, contrary to that constructed in [19,57], where the superflow (59) is not representable as a Hamiltonian superflow subject to some super-Poisson bracket $\{\mathsf{\Phi }(x,{\theta }_1),\mathsf{\Phi }(y,{\xi }_1)\}$, defined on the adjoint space ${\mathcal{K}}^{*}(1|1)$ for the local super-functionals $\mathsf{\Phi }(x,{\theta }_1)=u(x)+{\theta }_{1}v(x)$ and $\mathsf{\Phi }(y,{\xi }_1)=u(y)+{\xi }_{1}v(y)\in$ ${\mathcal{K}}^{*}(1|1)$ at $(x,{\theta }_1),(y,{\xi }_1)\in {\mathbb{S}}^{1|1}$.

Case $N=2$: Having put, by definition, ${\tilde{l}}_1(w;\lambda )=u(x,{\theta }_1)+{\theta }_{2}v(x,{\theta }_1)+\lambda \in \tilde{\mathcal{K}}{(1|2)}^{*}$, where $u\in C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_0^{(1)})$ and $v\in C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_1^{(1)})$, one ensues from the super-gradient relationship (35) the following compatible pair of super-Poisson matrix operators:

$$\vartheta =\left(\begin{array}{cc}0& \mathsf{\partial }\\ \mathsf{\partial }& 0\end{array}\right),\eta =\left(\begin{array}{cc}1/2(v-D_{{\vartheta }_1}^3)& -\mathsf{\partial }u+1/2(D_{{\theta }_1}u)D_{{\theta }_1}\\ -u\mathsf{\partial }-1/2u_x+1/2(D_{{\theta }_1}u)D_{{\theta }_1}& 1/2D_{{\theta }_1}^5-\mathsf{\partial }v-1/2v\mathsf{\partial }-1/2(D_{{\theta }_1}u)D_{{\theta }_1}\end{array}\right)$$

(60)

jointly with the countable hierarchy of the super-gradient covectors:

$$\begin{array}{cc}\hfill \mathrm{grad}{\gamma }_0[u,v]& =\left(\begin{array}{c}0\\ 1\end{array}\right),\mathrm{grad}{\gamma }_1[u,v]=\left(\begin{array}{c}v\\ u\end{array}\right),\mathrm{grad}{\gamma }_2[u,v]=\left(\begin{array}{c}-1/2\left(D_{{\theta }_1}{u}_x\right)+2uv\\ 1/2\left(D_{{\theta }_1}v\right)+u^2\end{array}\right),\hfill \\ \hfill \mathrm{grad}{\gamma }_3[u,v]& =\left(\begin{array}{c}-1/4v_{xx}+3/4v(D_{{\theta }_1}v)+3u^{2}v-3/2u(D_{{\theta }_1}{u}_x)-3/4u_x(D_{{\theta }_1}u)\\ -1/4u_{xx}+a^3+3/2u(D_{{\theta }_1}v)-3/4v(D_{{\theta }_1}u)\end{array}\right),...\hfill \end{array}$$

(61)

and so on. The corresponding invariants are given by superfunctionals

$$\begin{array}{cc}\hfill {\gamma }_0& ={\int }_0^{2\pi }dx\int d{\theta }_{1}v,{\gamma }_1={\int }_0^{2\pi }dx\int d{\theta }_{1}uv,{\gamma }_2={\int }_0^{2\pi }dx\int d{\theta }_1(-1/4u(D_{{\theta }_1}{u}_x)+1/4v(D_{{\theta }_1}v)+u^{2}v),\hfill \\ \hfill {\gamma }_3& ={\int }_0^{2\pi }dx\int d{\theta }_1(-1/4u{v}_{xx}+u^{3}v+3/4uv(D_{{\theta }_1}v)+3/4u{u}_x\left(D_{{\theta }_1}u\right)),...\hfill \end{array}$$

(62)

naturally generating a countable hierarchy of super-Hamiltonian flows. In particular, the so called Laberge–Mathieu superflow

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(\begin{array}{c}u\\ v\end{array}\right)=-\vartheta \mathrm{grad}{\gamma }_2[u,v]=\left.\begin{array}{c}-1/4u_{xxx}+3u^2{u}_x+3/4{\left((2D_{{\theta }_1}v)-v(D_{{\theta }_1}u)\right)}_x,\\ -1/4v_{xxx}+3/4{[v\left(D_{{\theta }_1}v\right)+4u^{2}v-2u(D_{{\theta }_1}{u}_x)-u_x(D_{{\theta }_1}u))}_x\end{array}\right\}$$

(63)

on the functional supermanifold $M^{1,1}\subset C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_0^{(1)}\times {\mathsf{\Lambda }}_1^{(1)})$ generalizes [15,16,24] the well-known classical modified Korteweg–de Vries dynamical system. What is worthy to remark here is that the superflow (63) is a true supersymmetric Hamiltonian system

$$\mathsf{\partial }\mathsf{\Phi }/\mathsf{\partial }t=\{{\gamma }_2,\mathsf{\Phi }\}$$

(64)

with respect to the following super-Poisson bracket

$$\begin{array}{c}\{\mathsf{\Phi }(x,{\theta }_1,{\theta }_2),\mathsf{\Phi }(y,{\xi }_1,{\xi }_2)\}=(D_{{\theta }_1}{D}_{{\theta }_2}\mathsf{\partial }+2\mathsf{\Phi }(x,{\theta }_1,{\theta }_2)\mathsf{\partial }-\langle D_{\theta }\mathsf{\Phi }(x,{\theta }_1,{\theta }_2)|D_{\theta }\rangle +\\ +2\mathsf{\Phi }{(x,{\theta }_1,{\theta }_2)}_x)\delta (x-y)({\xi }_2-{\theta }_2)({\xi }_1-{\theta }_1)\end{array}$$

(65)

and the Hamiltonian

$${\gamma }_2={\int }_0^{2\pi }dx\int d\theta \left(\mathsf{\Phi }{D}_{{\theta }_1}{D}_{{\theta }_2}\mathsf{\Phi }+1/3{\mathsf{\Phi }}^3\right),$$

(66)

being a smooth super-functional on the adjoint space $\mathcal{K}{(1|2)}^{*}$ jointly with local superfields $\mathsf{\Phi }(x,{\theta }_1,{\theta }_2)=u\left(x,{\theta }_1\right)+{\theta }_{2}v\left(x,{\theta }_1\right)$ and $\mathsf{\Phi }(y,{\xi }_1,{\xi }_2)=u\left(y,{\xi }_1\right)+{\xi }_{2}v\left(y,{\xi }_1\right)$ $\in {\mathcal{K}}^{*}(1|2)$ at $\left(x,{\theta }_1,{\theta }_2\right)$ and $(y,{\xi }_1,{\xi }_2)\in {\mathbb{S}}^{1|2}$, respectively.

Case $N=3$: We will choose the seed element (51) as ${\tilde{l}}_1(w;\lambda )=u(x,{\theta }_1,{\theta }_2){\theta }_3+v(x,{\theta }_1,{\theta }_2)+\lambda \in \tilde{\mathcal{K}}{(1|3)}^{*}$, where $u\in C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_0^{(3)})$ and $v\in C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_1^{(3)})$. As above, the corresponding Casimir super-gradients $\mathrm{grad}{\gamma }_j({\tilde{l}}_1),j\in {\mathbb{Z}}_{+},$ recurrently follow from (34) and equal expressions

$$\begin{array}{cc}\hfill \mathrm{grad}{\gamma }_0[u,v]& =\left(\begin{array}{c}u\\ 2v_x\end{array}\right),\hfill \\ \hfill \mathrm{grad}{\gamma }_1[u,v]& =\left(\begin{array}{c}{D}_{{\theta }_1}{D}_{{\theta }_2}u-\langle Dv|Dv\rangle +5/2u^2-2v{v}_x\\ 4D_{{\theta }_1}{D}_{{\theta }_2}{v}_x+2\langle Dv|Du\rangle +8u{v}_x+2u{v}_x\end{array}\right),\cdots \hfill \end{array}$$

(67)

and so on, where $\mathrm{grad}{\gamma }_j[u,v]={\left({\vartheta }^{-1}\eta \right)}^j\mathrm{grad}{\gamma }_0[u,v]$ for $j\in {\mathbb{Z}}_{+}$ and

$$\vartheta =\left(\begin{array}{cc}\mathsf{\partial }& 0\\ 0& 1/2\end{array}\right),\eta =\left(\begin{array}{cc}{D}_{{\theta }_1}{D}_{{\theta }_2}\mathsf{\partial }-\langle Du|D\rangle +2\mathsf{\partial }u& -\mathsf{\partial }v-\langle Dv|D\rangle \\ v_x+\mathsf{\partial }v+\langle Dv|D\rangle & D_{{\theta }_1}{D}_{{\theta }_2}+u\end{array}\right)$$

(68)

where we denoted, for brevity, that the super-gradient $(D_{{\theta }_1},D_{{\theta }_2}):=D$ is the corresponding compatible Poisson structures on the functional supermanifold $M^{1,1}\subset C^{\infty }({\mathbb{S}}^1;{\mathsf{\Lambda }}_0^{(1)}\times {\mathsf{\Lambda }}_1^{(1)})$. The related countable hierarchy of superfunctionals

$$\begin{array}{cc}\hfill {\gamma }_0& ={\int }_0^{2\pi }dx\int d\theta (u^2/2+v{v}_x),{\gamma }_1={\int }_0^{2\pi }dx\int d\theta (3/2D_{{\theta }_1}{D}_{{\theta }_2}u+5/2u^3+6uv{v}_x-\hfill \\ \hfill & -u\langle Dv|Dv\rangle +2v\langle Dv|Du\rangle +6v{D}_{{\theta }_1}{D}_{{\theta }_2}{v}_x),...\hfill \end{array}$$

(69)

is invariant subject to the following suitably integrable generating Hamiltonian superflow

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(\begin{array}{c}u\\ v\end{array}\right)=-\vartheta \mathrm{grad}{\gamma }_1[u,v]=\left.\begin{array}{c}{D}_{{\theta }_1}{D}_{{\theta }_2}{u}_x+5u{u}_x-2v{v}_{xx}-2\langle Dv|D{v}_x\rangle \\ 2D_{{\theta }_1}{D}_{{\theta }_2}{v}_x+\langle Dv|Du\rangle +4u{v}_x+u_{x}v\end{array}\right\}$$

(70)

on the functional supermanifold $M^{1,1}$. Contrary to the case of $N=2$ considered before, the above-constructed Hamiltonian superflow (70) on the functional supermanifold $M^{1,1}$ is not supersymmetric.

Nonetheless, the linear integro-differential super-Sturm–Liouville-type spectral problems (44)–(46) constructed before make it possible for the linear in $\lambda \in \mathbb{C}$ seed element ${\tilde{l}}_1(w;\lambda )=w_0\lambda +w_1\in \tilde{\mathcal{K}}{(1|N)}^{*},N=\overline{1,3},$ to be generalized to super-differential operator expressions of both arbitrary order and fractional structure, allowing one to generate countable hierarchies of new integrable supersymmetric Hamiltonian systems on the corresponding supermanifolds $M^{k,m}\subset C^{\infty }({\mathbb{S}}^{1|N};{\left({\mathsf{\Lambda }}_0^{(N)}\right)}^k\times {\left({\mathsf{\Lambda }}_1^{(N)}\right)}^m),$ $N=\overline{1,3},$ for some $k,m\in \mathbb{N}$.

4. Super-Differential Operator-Valued Coadjoint Orbits and the Related Supersymmetric Hamiltonian Systems

Consider the simplest case, where the superpotential (47) is linear with respect to the spectral parameter $\lambda \in \mathbb{C}$, where $w_j:{\mathbb{S}}^{1|N}\to {\mathsf{\Lambda }}_{N\mathrm{mod}(2)}^{(N)}$, $N=\overline{1,3}, j\in \overline{0,1}$. Then, we can follow reasoning from [15] based on construction of the associative superalgebras $A(1|N),N=\overline{1,3},$ of pseudo-superdifferential operators, containing the corresponding seed operator expressions $L^{(N)}$ for $N=\overline{1,3}$:

$$\begin{array}{c}{L}^{(1)}\in A(1|1):=\{\sum _{j\le n}(a_{0,j}+a_{1,j}{D}_1){\mathsf{\partial }}^j:\\ a_{k,j}\in C^{\infty }({\mathbb{S}}^{1|1};{\mathsf{\Lambda }}^{(1)}),k=\overline{0,1},n\in \mathbb{Z}\},\\ \\ L^{(2)}\in A(1|2):=\{\sum _{j\le n}(a_{0,j}+a_{1,j}{D}_1+a_{2,j}{D}_2+a_{12,j}{D}_1{D}_2){\mathsf{\partial }}^j\\ :a_{k,j},a_{12,j}\in C^{\infty }({\mathbb{S}}^{1|2};{\mathsf{\Lambda }}^{(2)}),k=\overline{0,2},n\in \mathbb{Z}\}\end{array}$$

(71)

and

$$\begin{array}{c}{L}^{(3)}\in A(1|3):=\{\sum _{j\le n}(a_{0,j}+a_{1,j}{D}_1+a_{2,j}{D}_2+a_{3,j}{D}_3+\\ +a_{12,j}{D}_1{D}_2+a_{13,j}{D}_1{D}_3+a_{23,j}{D}_2{D}_3+a_{123,j}{D}_1{D}_2{D}_3){\mathsf{\partial }}^j:\\ a_{123,j},a_{ks,j},a_{k,j}\in C^{\infty }({\mathbb{S}}^{1,3};{\mathsf{\Lambda }}^{(3)}),k<s=\overline{0,3},n\in \mathbb{Z}\},\end{array}$$

(72)

where we put, by definition, $D_j:=D_{{\theta }_j}=\mathsf{\partial }/\mathsf{\partial }{\theta }_j+{\theta }_j\mathsf{\partial }/\mathsf{\partial }x,j=\overline{1,3}$. These associative superalgebras $A(1|N),N=\overline{1,3},$ can be endowed with the standard Lie super-commutator $[·,·]$, naturally transforming them into the operator Lie superalgebras $\mathcal{L}(1|N),N=\overline{1,3},$ characterized by the following important lemma.

Lemma 1. 

Let the operator Lie superalgebras $\mathcal{L}(1|N),N=\overline{1,3},$ be equipped with trace-operations $\mathrm{tr}:\mathcal{L}(1|N)\to {\mathsf{\Lambda }}_0^{(N)},$ $N=\overline{1,3},$ defined, respectively, as

$$\mathrm{tr}\left(A\right):={\int }_0^{2\pi }dx\int d{\theta }_1{\mathrm{res}}_{D_1{\mathsf{\partial }}^{-1}}A(x,{\theta }_1)$$

(73)

for the Lie superalgebra $\mathcal{L}(1|1)$,

$$\mathrm{tr}\left(A\right):={\int }_0^{2\pi }dx\int \int d{\theta }_{1}d{\theta }_2{\mathrm{res}}_{D_1{D}_2{\mathsf{\partial }}^{-1}}A(x,{\theta }_1,{\theta }_2)$$

(74)

for the Lie superalgebra $\mathcal{L}(1|2)$, and

$$\mathrm{tr}\left(A\right):={\int }_0^{2\pi }dx\int \int d{\theta }_{1}d{\theta }_{2}d{\theta }_3{\mathrm{res}}_{D_1{D}_2{D}_3{\mathsf{\partial }}^{-1}}A(x,{\theta }_1,{\theta }_2,{\theta }_3)$$

(75)

for the Lie superalgebra $\mathcal{L}(1|3)$. Then, the operator Lie superalgebras $\mathcal{L}(1|N),N=\overline{1,3},$ are metrized subject to the corresponding supersymmetric trace-type bilinear form $(·|·):\mathcal{L}(1|N)\times \mathcal{L}(1|N)\to {\mathsf{\Lambda }}_0^{(N)}$, where, for any $A,B\in \mathcal{L}(1|N),N=\overline{1,3},$

$$(A|B):=\mathrm{tr}\left(AB\right),$$

(76)

that is,

$$([A,B]|C)=(A|[B,C])$$

(77)

for arbitrary uniform elements $A,B$, and $C\in \mathcal{L}(1|N),N=\overline{1,3}$. Moreover, subject to these trace-operations, the Lie superalgebras $\mathcal{L}(1|N),N=\overline{1,3},$ can be identified, respectively, with the adjoint spaces $\mathcal{L}{(1|N)}^{*},N=\overline{1,3}$; that is, $\mathcal{L}{(1|N)}^{*}\simeq \mathcal{L}(1|N),N=\overline{1,3}$.

We are now interested in constructing integrable supersymmetric Hamiltonian systems within the classical Adler–Kostant–Symes Lie-algebraic scheme [42,44,45,46,48,49], related to coadjoint orbits of the Lie superalgebras $\mathcal{L}(1|N),N=\overline{1,3},$ at a specially chosen element $L\in \mathcal{L}{(1|N)}^{*}\simeq \mathcal{L}(1|N)$ of the adjoint spaces $\mathcal{L}{(1|N)}^{*},N=\overline{1,3},$ identified, respectively, with the Lie superalgebras $\mathcal{L}(1|N),N=\overline{1,3}$. As the first step, we need to split each Lie superalgebra $\mathcal{L}(1|N),N=\overline{1,3},$ as the direct sum of two Lie subalgebras: $\mathcal{L}(1|N)={\mathcal{L}}_{+}(1|N)\oplus {\mathcal{L}}_{-}(1|N)$, where, by definition,

$$\begin{array}{cc}\hfill {\mathcal{L}}_{+}(1|N)& :=\{A=\sum _{j=0}^{\ll \infty }(a_{0,j}+a_{1,j}{D}_1+a_{2,j}{D}_2+a_{3,j}{D}_1{D}_2){\mathsf{\partial }}^j:a_{k,j}\in C^{\infty }({\mathbb{S}}^{1|2};{\mathsf{\Lambda }}^{(N)}),k=\overline{0,3},j\in {\mathbb{Z}}_{+}\},\hfill \\ \hfill {\mathcal{L}}_{-}({\mathbb{S}}^{1|2})& :=\{A=\sum _{j=1}^{\infty }(a_{0,j}+a_{1,j}{D}_1+a_{2,j}{D}_2+a_{3,j}{D}_1{D}_2){\mathsf{\partial }}^{-j}:a_{k,j}\in C^{\infty }({\mathbb{S}}^{1|2};{\mathsf{\Lambda }}^{(N)}),k=\overline{0,3},j\in \mathbb{N}\},\hfill \end{array}$$

(78)

allowing us to define [42,44,45,49] on $\mathcal{L}(1|N)$ a second Lie commutator structure, the so-called $R-$structure:

$${[A,B]}_R:=[RA,B]+[A,RB]$$

(79)

for any $A,B\in \mathcal{L}(1|N)$, where $R:=\left(P_{+}-P_{-}\right)/2,P_{\pm }\mathcal{L}(1|N):={\mathcal{L}}_{\pm }(1|N)\subset \mathcal{L}(1|N)$ are the corresponding projectors on ${\mathcal{L}}_{\pm }(1|N)\subset \mathcal{L}(1|N)$, satisfying the Jacobi identity. Within the above-mentioned AKS scheme, a countable hierarchy of smooth Casimir invariants ${\gamma }_j\in I(\mathcal{L}{(1|N)}^{*}),j\in \mathbb{N},$ of the Lie superalgebra $\mathcal{L}(1|N)$, satisfying the determining relationships

$$a{d}_{\mathrm{grad}{\gamma }_j(L)}^{*}L=0\simeq [\mathrm{grad}{\gamma }_j(L),L]=0$$

(80)

along coadjoint orbits at a seed element $L\in \mathcal{L}{(1|N)}^{*}$, generates a countable hierarchy of evolution systems commuting to each Lax type, respectively

$$\mathsf{\partial }L/\mathsf{\partial }{t}_j=[P_{+}\mathrm{grad}{\gamma }_j(L),L].$$

(81)

The latter proves to be equivalent to a countable hierarchy of completely integrable super-Hamiltonian systems on a smooth functional supermanifold $M^{k,m},$ $k,m\in N$, suitably related with coefficients of the chosen orbit element $L\in \mathcal{L}{(1|N)}^{*}$. This hierarchy can be naturally extended by means of replacing the Lie superalgebra $\mathcal{L}(1|N)$ by the centrally extended Lie superalgebra $\widehat{\mathcal{L}}(1|N):=\tilde{\mathcal{L}}{(1|N)|}_y\oplus \mathbb{C}$, where $\tilde{\mathcal{L}}(1|N)=$ ${\displaystyle \prod _{y\in {\mathbb{S}}^1}}{\mathcal{L}(1|N)|}_y$, and, for arbitrary $(\tilde{A},\alpha ),(\tilde{B},\beta )\in \widehat{\mathcal{L}}(1|N)$, the super-commutator is given by the expression

$$[(\tilde{A},\alpha ),(\tilde{B},\beta )]=({[\tilde{A},\tilde{B}]}_R,{\omega }_1(\tilde{A},\tilde{B}))$$

(82)

with the classical Maurer–Cartan [49] cocycle:

$${\omega }_1(\tilde{A},\tilde{B})={\int }_{{\mathbb{S}}^1}\mathrm{tr}(\tilde{A}(x,\theta ;y)d_y\tilde{B}(x,\theta ;y).$$

(83)

The bracket (82) makes it possible to construct the canonical Lie–Poisson bracket

$${\{\gamma ,\mu \}}_{Lie-P}:=(\tilde{L}|[\mathrm{grad}\gamma (\tilde{L}),\mathrm{grad}\mu (\tilde{L})])+(\mathrm{grad}\gamma (\tilde{L})|{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}\mu (\tilde{L}))$$

(84)

on the adjoint superspace $\widehat{\mathcal{L}}(1|N)$ for arbitrary smooth functionals $\gamma ,\mu :\widehat{\mathcal{L}}{(1|N)}^{*}\to \mathbb{C}$, subject to which the related Casimir invariants ${\gamma }_j\in I(\widehat{\mathcal{L}}{(1|N)}^{*}),j\in \mathbb{N},$ satisfy the determining relationships

$$a{d}_{\mathrm{grad}{\gamma }_j(\tilde{L})}^{*}\tilde{L}-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}{\gamma }_j(\tilde{L})=0\simeq [\mathrm{grad}{\gamma }_j(\tilde{L}),\tilde{L}]+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}{\gamma }_j(\tilde{L})=0$$

(85)

along coadjoint orbits at a seed element $(\tilde{L},1)\in \widehat{\mathcal{L}}{(1|N)}^{*}\simeq \tilde{\mathcal{L}}{(1|N)}^{*}\oplus \mathbb{C}$. Here, as was performed above, the adjoint space $\tilde{\mathcal{L}}{(1|N)}^{*}\simeq \tilde{\mathcal{L}}(1|N)$ with respect to the following slightly modified nondegenerate and symmetric trace-operation on the Lie superalgebra $\tilde{\mathcal{L}}(1|N)$:

$$(\tilde{A}|\tilde{B}):=\mathrm{Tr}(\tilde{A}\tilde{B})={\int }_{{\mathbb{S}}^1}dy\mathrm{tr}(\tilde{A}\tilde{B}),$$

(86)

defined for any $\tilde{A},\tilde{B}\in \tilde{\mathcal{L}}(1|N)$, subject to which the Lie superalgebra $\tilde{\mathcal{L}}(1|N)$ persists to be metrized; that is,

$$([\tilde{A},\tilde{B}]|\tilde{C})=(\tilde{A}|[\tilde{B},\tilde{C}])$$

(87)

for arbitrary uniform elements $\tilde{A},\tilde{B}$, and $\tilde{C}\in \tilde{\mathcal{L}}(1|N)$. The centrally extended Lie superalgebra $\widehat{\mathcal{L}}(1|N)$ can be similarly endowed with the R-structure

$${[(\tilde{A},\alpha ),(\tilde{B},\beta )]}_R=\left({[\tilde{A},\tilde{B}]}_R,{\omega }_1(R\tilde{A},\tilde{B})+{\omega }_1(\tilde{A},R\tilde{B})\right),$$

(88)

subject to which the above-constructed Casimir invariants ${\gamma }_j\in I(\widehat{\mathcal{L}}{(1|N)}^{*}),j\in \mathbb{N}$, as follows [41,42,45,46,47,49] from the Adler–Kostant–Symes scheme, are commuting to each other with respect to the corresponding Lie–Poisson bracket

$$\begin{array}{cc}\hfill \{\gamma ,\mu \}& :=(\tilde{L}|[{[\mathrm{grad}\gamma (\tilde{L}),\mathrm{grad}\mu (\tilde{L})]}_R+(R\mathrm{grad}\gamma (\tilde{L})|{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}\mu (\tilde{L}))+\hfill \\ \hfill & +(\mathrm{grad}\gamma (\tilde{L})|{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}R\mathrm{grad}\mu (\tilde{L})),\hfill \end{array}$$

(89)

defined for arbitrary smooth functionals $\gamma ,\mu :\widehat{\mathcal{L}}{(1|N)}^{*}\to \mathbb{C},$ that is $\{{\gamma }_j,{\gamma }_k\}=0$ for all $j,k\in \mathbb{N}$. In particular, any smooth functional $\gamma :\widehat{\mathcal{L}}{(1|N)}^{*}\to \mathbb{C}$ generates the following Hamiltonian superflow on an operator element $\tilde{L}\in \widehat{\mathcal{L}}{(1|N)}^{*}$:

$$\mathsf{\partial }\tilde{L}/\mathsf{\partial }t=[P_{+}\mathrm{grad}\gamma (\tilde{L}),\tilde{L}-\mathsf{\partial }/\mathsf{\partial }y]-P_{+}[\mathrm{grad}\gamma (\tilde{L}),\tilde{L}-\mathsf{\partial }/\mathsf{\partial }y]$$

(90)

with respect to the temporal parameter $t\in \mathbb{C}$. In particular, the Casimir invariants ${\gamma }_j\in I(\widehat{\mathcal{L}}{(1|N)}^{*}),$ $j\in \mathbb{N},$ generate with respect to the Lie–Poisson bracket (89) the related countable hierarchy of Hamiltonian flows commuting to each other:

$${\displaystyle \frac{\mathsf{\partial }\tilde{L}}{\mathsf{\partial }{t}_j}}=[P_{+}\mathrm{grad}{\gamma }_j(\tilde{L}),\tilde{L}]+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\left(P_{+}\mathrm{grad}{\gamma }_j(\tilde{L})\right)$$

(91)

for all $j\in \mathbb{N}$. For the flows (91) to be written explicitly for a given a priori seed element $\tilde{L}\in \tilde{\mathcal{L}}{(1|N)}^{*}$, one needs to construct algorithmically the countable hierarchy of the Casimir functionals ${\gamma }_j\in I(\widehat{\mathcal{L}}{(1|N)}^{*}),j\in \mathbb{N},$ making use of the scheme devised before in [44]. In particular, the following lemma, generalizing those in [21,35,44], holds.

Lemma 2. 

Any operator expression $\tilde{L}\in \tilde{\mathcal{L}}{(1|N)}^{*}$ can be transformed by means of the gauge transformation $\tilde{T}\in exp\left(\tilde{\mathcal{L}}{(1|N)}_{\_}\right)$ to the operator expression $\overline{L}\in \tilde{\mathcal{L}}{(1|N)}^{*}$, where

$$(\mathsf{\partial }/\mathsf{\partial }y-\tilde{L})\circ \tilde{T}=\tilde{T}\circ (\mathsf{\partial }/\mathsf{\partial }y-\overline{L})$$

(92)

and $\mathsf{\partial }\overline{L}/\mathsf{\partial }x=0$ for $x\in {\mathbb{S}}^1$ and all $y\in {\mathbb{S}}^1$. Moreover, the functionals

$${\gamma }_n:={\int }_{{\mathbb{S}}^1}{\mathrm{res}}_{D_{\theta }{\mathsf{\partial }}^{-1}}\left({\mathsf{\partial }}^n\overline{L}(y)\right)dy,$$

(93)

where $n\ge -\mathrm{ord}\tilde{L}$ and $D_{\theta }:=D_{{\theta }_1}$ for $N=1,D_{\theta }:=D_{{\theta }_1}{D}_{{\theta }_2}$ for $N=2$, and $D_{\theta }:=D_{{\theta }_1}{D}_{{\theta }_2}{D}_{{\theta }_3}$ for $N=3$ are invariant with respect to all Hamiltonian flows (91) and satisfy the Casimir determining relationships (85).

Proof. 

It is enough to check that the gradient elements $\mathrm{grad}{\gamma }_n(\tilde{L})\in \tilde{\mathcal{L}}(1|N),n\ge -\mathrm{ord}\tilde{L},$ satisfy the Casimir determining relationships (85). Since, for all integers $n\ge -\mathrm{ord}\tilde{L}$, owing to the definition (92),

$$\delta {\gamma }_n=\mathrm{Tr}\left({\mathsf{\partial }}^n\delta \overline{L}(y)\right)=\mathrm{Tr}\left(\tilde{T}{\mathsf{\partial }}^n{\tilde{T}}^{-1}\delta \tilde{L}(y)\right)=(\mathrm{grad}{\gamma }_n(\tilde{L})|\delta \tilde{L}),$$

(94)

one easily obtains that the elements $\mathrm{grad}{\gamma }_n(\tilde{L})=\tilde{T}{\mathsf{\partial }}^n{\tilde{T}}^{-1}\in \tilde{\mathcal{L}}(1|N)$ satisfy the following operator commutators:

$$\begin{array}{c}[\mathrm{grad}{\gamma }_n(\tilde{L}),\tilde{L}]=[\tilde{T}{\mathsf{\partial }}^n{\tilde{T}}^{-1},\mathsf{\partial }\tilde{T}/\mathsf{\partial }y{\tilde{T}}^{-1}+\tilde{T}\overline{L}{\tilde{T}}^{-1}]=\\ =\tilde{T}{\mathsf{\partial }}^n{\tilde{T}}^{-1}\mathsf{\partial }\tilde{T}/\mathsf{\partial }y{\tilde{T}}^{-1}-\mathsf{\partial }\tilde{T}/\mathsf{\partial }y{\mathsf{\partial }}^n{\tilde{T}}^{-1}+\tilde{T}[{\mathsf{\partial }}^n,\overline{L}]{\tilde{T}}^{-1}=\\ =-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\left(\tilde{T}{\mathsf{\partial }}^n{\tilde{T}}^{-1}\right)=-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\left(\mathrm{grad}{\gamma }_n(\tilde{L})\right),\end{array}$$

(95)

being completely equivalent to the Casimir determining relationships (85), proving the lemma. □

Remark 1. 

It is worth mentioning that the operator expression (92) makes it possible to construct the determining gauge transformation $\tilde{T}\in exp\left(\tilde{\mathcal{L}}{(1|N)}_{\_}\right)$ exactly by means of a countable hierarchy of recurrent operator relationships, thus implying that the countable hierarchy of the Casimir functionals ${\gamma }_j\in I(\widehat{\mathcal{L}}{(1|N)}^{*}), j\in \mathbb{N},$ is simultaneously constructed too.

Example 2. 

Let $N=1$ and choose the seed element $\tilde{L}\in \tilde{\mathcal{L}}{(1|1)}^{*}$ as

$$\tilde{L}=D_{\theta }^4+\mathsf{\Phi }{D}_{\theta },$$

(96)

where $\mathsf{\Phi }:=\mathsf{\Phi }(x,{\theta }_1;y)\in {\mathsf{\Lambda }}_1^{(1)},(x,\theta ;y)\in {\mathbb{S}}^{1|1}\times {\mathbb{S}}^1$. Having assumed, for brevity, that $\mathsf{\partial }\mathsf{\Phi }/\mathsf{\partial }y=0,y\in {\mathbb{S}}^1$, one easily constructs a countable hierarchy of supersymmetric invariants ${\gamma }_j=\mathrm{Tr}{(D_{\theta }^4+\mathsf{\Phi }{D}_{\theta })}^{j/2+1},j\in \mathbb{N},$ in particular,

$${\gamma }_1=\int dx\int d\theta \mathsf{\Phi }{\mathsf{\Phi }}_x,{\gamma }_2=\int dx\int d\theta \mathsf{\Phi }({\mathsf{\Phi }}_{xxx}+4{\mathsf{\Phi }}_x{D}_{\theta }\mathsf{\Phi }),...$$

(97)

and so on, generating integrable superflows on a functional supermanifold $M^{1,1}\subset C({\mathbb{S}}^{1|1};{\mathsf{\Lambda }}_1^{(1)})$ endowed with two compatible Poisson structures:

$$\{\mathsf{\Phi }(x,\theta ),\mathsf{\Phi }(x^{\prime },{\theta }^{\prime }){\}}_{\vartheta }=D_{\theta }^{-1}({\mathsf{\partial }}^3+2\mathsf{\partial }\left(D_{\theta }\mathsf{\Phi }\right)+2\left(D_{\theta }\mathsf{\Phi }\right)\mathsf{\partial })D_{\theta }^{-1}\delta (x-x^{\prime })(\theta -{\theta }^{\prime })$$

(98)

and

$$\{\mathsf{\Phi }(x,\theta ),\mathsf{\Phi }(x^{\prime },{\theta }^{\prime }){\}}_{\eta }=D_{\theta }^{-1}\mathsf{\partial }{D}_{\theta }^{-1}\delta (x-x^{\prime })(\theta -{\theta }^{\prime }),$$

(99)

where $(x,\theta )$ and $(x^{\prime },{\theta }^{\prime })\in {\mathbb{S}}^{1|1}$. For the case ${\gamma }_2\in I($ $\widehat{\mathcal{L}}{(1|1)}^{*})$, the resulting superflow ${\tilde{L}}_t=4[\mathrm{grad}{\gamma }_2(\tilde{L}),\tilde{L}]$ on $\widehat{\mathcal{L}}{(1|1)}^{*}$ is equivalent to the super-Hamiltonian system

$${\mathsf{\Phi }}_t=({\mathsf{\Phi }}_{xx}+3\mathsf{\Phi }{D}_{\theta }\mathsf{\Phi }{)}_x$$

(100)

of the well-known [6,21,22] Korteweg–de Vries-type superflow.

Example 3. 

Take $N=2$ and let the seed element $\tilde{L}\in \tilde{\mathcal{L}}{(1|2)}^{*}$ be given by the following expression:

$$\begin{array}{cc}\hfill \tilde{L}& ={\mathsf{\partial }}^2+\mathsf{\Phi }{D}_{{\theta }_1}{D}_{{\theta }_{2n}}+a(D_{{\theta }_1}\mathsf{\Phi }+D_{{\theta }_2}\mathsf{\Phi })D_{{\theta }_1}-\hfill \\ \hfill & -a(D_{{\theta }_1}\mathsf{\Phi }+D_{{\theta }_2}\mathsf{\Phi })D_{{\theta }_2}+D_{{\theta }_1}{D}_{{\theta }_2}\mathsf{\Phi }-{\mathsf{\Phi }}^2,\hfill \end{array}$$

(101)

where $a=-2,1,4$ and $\mathsf{\Phi }:=\mathsf{\Phi }(x,\theta ;y)\in {\mathsf{\Lambda }}_0^{(2)},$ $(x,\theta ;y)\in {\mathbb{S}}^{1|2}\times {\mathbb{S}}^1$. Having assumed that $\mathsf{\partial }\mathsf{\Phi }/\mathsf{\partial }y=0,y\in {\mathbb{S}}^1$, one easily derives the suitably reduced Casimir functionals:

$$\begin{array}{cc}\hfill {\gamma }_1& =\int dx\int d\theta \mathsf{\Phi }(x,\theta ),{\gamma }_2=\int dx\int d\theta {\mathsf{\Phi }}^2(x,\theta ),\hfill \\ \hfill {\gamma }_3& =\int dx\int d\theta (\mathsf{\Phi }{(x,\theta )}^3+3/a\mathsf{\Phi }(x,\theta )D_{{\theta }_1}{D}_{{\theta }_2}\mathsf{\Phi }),\hfill \\ \hfill {\gamma }_4& =\int dx\int d\theta (\mathsf{\Phi }{(x,\theta )}^4+1/2{\mathsf{\Phi }}_x{(x,\theta )}^2+3/2\mathsf{\Phi }{(x,\theta )}^2{D}_{{\theta }_1}{D}_{{\theta }_2}\mathsf{\Phi }(x,\theta )),...\hfill \end{array}$$

(102)

and so on. In particular, the Hamiltonian functional $H={\gamma }_3\in I(\tilde{\mathcal{L}}{(1|2)}^{*})$ generates the following supersymmetric Korteweg–de Vries-type Hamiltonian system:

$${\mathsf{\Phi }}_t=-{\mathsf{\Phi }}_{xxx}+3(\mathsf{\Phi }{D}_{{\theta }_1}{D}_{{\theta }_2}\mathsf{\Phi }{)}_x+1/2(a-1){(D_{{\theta }_1}{D}_{{\theta }_2}{\mathsf{\Phi }}^2)}_x+3a{\mathsf{\Phi }}^2{\mathsf{\Phi }}_x,$$

(103)

invariant with respect to the generating supersymmetry flows

$${\mathsf{\Phi }}_{{\tau }_1}=D_{{\theta }_1}\mathsf{\Phi },{\mathsf{\Phi }}_{{\tau }_2}=D_{{\theta }_2}\mathsf{\Phi }.$$

(104)

The corresponding higher supersymmetric dynamical systems ${\mathsf{\Phi }}_{t_j}=\{{\gamma }_j,\mathsf{\Phi }\},j\in \mathbb{N},$ with respect to the corresponding Poisson structure

$$\begin{array}{cc}\hfill \{\mathsf{\Phi }(x,\theta ),\mathsf{\Phi }(x,\theta ){\}}_{\eta }& =(D_{{\theta }_1}{D}_{{\theta }_2}\mathsf{\partial }+2\mathsf{\partial }\mathsf{\Phi }+2\mathsf{\Phi }\mathsf{\partial }-D_{{\theta }_2}\mathsf{\Phi }{D}_{{\theta }_2}-\hfill \\ \hfill & -D_{{\theta }_2}\mathsf{\Phi }{D}_{{\theta }_2})({\theta }_1-{\theta }_1^{\prime })({\theta }_2-{\theta }_2^{\prime })\delta (x-x^{\prime })\hfill \end{array}$$

(105)

give rise to the $N=2$ super-Korteweg-de Vries-type [6,15,16] superflows commuting to each other on a smooth functional supermanifold $M^{1,1}\subset C({\mathbb{S}}^{1|2}\times {\mathbb{S}}^1;{\mathsf{\Lambda }}_1^{(2)})$. If the dependence on the variable $y\in {\mathbb{S}}^1$ is saved, the seed element (101) gives rise to a semi-supersymmetric version of the Kadomtsev–Petviashvily dynamical system [55,58,59] on a smooth functional supermanifold $M^{2,2}\subset C({\mathbb{S}}^{1|2}\times {\mathbb{S}}^1;$ ${\mathsf{\Lambda }}_0^{(2)})$.

In the case of generally chosen seed elements $\tilde{L}\in \tilde{\mathcal{L}}{(1|N)}^{*},N=\overline{1,3},$ the resulting Hamiltonian dynamical superflows (91) are not a priori semi-supersymmetric, as this property is guaranteed by the additional constraint to be invariant with respect to the supersymmetry generators $D_{{\theta }_j}=\mathsf{\partial }/\mathsf{\partial }{\theta }_j+{\theta }_j\mathsf{\partial }/\mathsf{\partial }x,j=\overline{1,3}$.

5. The Factorized Semi-Supersymmetric Hamiltonian Systems and Related Poisson Structures

We will now be interested by the Casimir invariant functionals ${\gamma }_j\in I(\tilde{\mathcal{L}}{(1|N)}^{*}),j\in \mathbb{N},$ for $N=\overline{1,3},$ satisfying the determining relationships

$$[\mathrm{grad}{\gamma }_j(\tilde{L}),\tilde{L}]+\mathsf{\partial }/\mathsf{\partial }y\mathrm{grad}{\gamma }_j(\tilde{L})=0,$$

(106)

and calculated at the following rationally factorized super-pseudo-differential operator element

$$\tilde{L}(x,\theta ):={\tilde{F}}_n{(x,\theta ;y)}^{-1}\circ {\tilde{Q}}_{n+p}(x,\theta ;y)$$

(107)

at $(x,\theta ;y)\in {\mathbb{S}}^{1|N}\times {\mathbb{S}}^1$, where, by definition, the operators

$$\begin{array}{cc}\hfill {\tilde{F}}_n(x,\theta ;y)& :=\sum _{j=\overline{0,n}}{f}_j(x,\theta ;y){\mathsf{\partial }}^j,\hfill \\ \hfill {\tilde{Q}}_{n+p}(x,{\theta }^{\prime }y)& :=\sum _{j=\overline{0,n+p}}{q}_j(x,\theta ;y){\mathsf{\partial }}^j\hfill \end{array}$$

(108)

are some super-differential polynomial expressions of $exp\left(\tilde{\mathcal{L}}{(1|N)}_{+}\right)$; that is, ${\tilde{F}}_n,{\tilde{Q}}_{n+p}\in exp\left(\tilde{\mathcal{L}}{(1|N)}_{+}\right)$ for fixed integers n and $p\in \mathbb{N}$. The above-constructed Casimir functionals ${\gamma }_j\in I(\tilde{\mathcal{L}}{(1|N)}^{*}),j\in \mathbb{N},$ naturally generate a countable hierarchy of Lax-integrable Hamiltonian superflows

$$\mathsf{\partial }\tilde{L}/\mathsf{\partial }{t}_j=[\mathrm{grad}{\gamma }_j{(\tilde{L})}_{+},\tilde{L}]+\mathsf{\partial }/\mathsf{\partial }y\mathrm{grad}{\gamma }_j{(\tilde{L})}_{+}$$

(109)

along the seed element $\tilde{L}={\tilde{F}}_n^{-1}{\tilde{Q}}_{n+p}\in \tilde{\mathcal{L}}(1|N)$ and

$$\mathsf{\partial }{\tilde{L}}^{\prime }/\mathsf{\partial }{t}_j=[\mathrm{grad}{\gamma }_j{({\tilde{L}}^{\prime })}_{+},{\tilde{L}}^{\prime }]+\mathsf{\partial }/\mathsf{\partial }y\mathrm{grad}{\gamma }_j{({\tilde{L}}^{\prime })}_{+}$$

(110)

along the associated seed element

$${\tilde{L}}^{\prime }(x,\theta ):={\tilde{Q}}_{n+p}(x,\theta ;y)\circ {\tilde{F}}_n{(x,\theta ;y)}^{-1}$$

(111)

at $(x,\theta ;y)\in {\mathbb{S}}^{1|N}\times {\mathbb{S}}^1$, where we denoted ${(...)}_{\pm }:=P_{\pm }(...)$, respectively, as the projections on the corresponding Lie super-subalgebras $\tilde{\mathcal{L}}{(1|N)}_{\pm }$.

Problem 1. 

The following problem [10,32,35,60,61,62] arises: how to construct the corresponding supersymmetric Hamiltonian systems on the operator elements ${\tilde{F}}_n$ and ${\tilde{Q}}_{n+p}\in exp(\tilde{\mathcal{L}}{(1|N)}_{+}),$ which will possess an infinite hierarchy of functional invariants commuting to each other and will be suitably integrable.

To specify the searched-for factorized superflows on operator factors ${\tilde{F}}_n$ and ${\tilde{Q}}_{n+p}\in$ $exp\left(\tilde{\mathcal{L}}{(1|N)}_{+}\right)$, we will analyze preliminarily the evolution of the superflows, generated by the Hamiltonian superflows (109) and (110). Namely, we can observe that the following evolution systems

$$\begin{array}{cc}\hfill \mathsf{\partial }\tilde{L}/\mathsf{\partial }{t}_j& =[\mathrm{grad}{\gamma }_j{(\tilde{L})}_{+},\tilde{L}]+\mathsf{\partial }/\mathsf{\partial }y\mathrm{grad}{\gamma }_j{(\tilde{L})}_{+}\hfill \\ \hfill & \hfill \\ \hfill \mathsf{\partial }{\tilde{f}}_j/\mathsf{\partial }t& =\mathrm{grad}{\gamma }_j{(\tilde{L})}_{+}{\tilde{f}}_j,\mathsf{\partial }{\tilde{f}}_j^{*}/\mathsf{\partial }{t}_j=-\mathrm{grad}{\gamma }_j{(\tilde{L})}_{+}^{*}{\tilde{f}}_j^{*},\hfill \end{array}$$

(112)

and

$$\mathsf{\partial }{\tilde{f}}_j/\mathsf{\partial }y=\tilde{L}{\tilde{f}}_j,\mathsf{\partial }{f}_j^{*}/\mathsf{\partial }y=-{\tilde{L}}^{*}{\tilde{f}}_j^{*},$$

(113)

are compatible on the space of functions ${\tilde{f}}_j$ and ${\tilde{f}}_j^{*}$ $\in L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0),j\in \mathbb{N},$ if the functionals ${\gamma }_j\in I(\tilde{\mathcal{L}}{(1|N)}^{*}),j\in \mathbb{N},$ are Casimir; that is, the operator relationships

$$[\mathrm{grad}{\gamma }_j(\tilde{L}),\tilde{L}]+\mathsf{\partial }/\mathsf{\partial }y\mathrm{grad}\gamma (\tilde{L})=0,$$

(114)

for $\tilde{L}\in \tilde{\mathcal{L}}{(1|N)}^{*}$ hold. It is easy to check that the combined systems of Equation (112) are not Hamiltonian with respect to the canonical tensor Poisson structure

$$\widehat{\mathsf{\Omega }}\otimes \widehat{J}=\left(\begin{array}{ccc}[P_{+}(...),\widehat{L}-\mathsf{\partial }/\mathsf{\partial }y]-P_{+}[(...),\widehat{L}-\mathsf{\partial }/\mathsf{\partial }y]& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right)$$

(115)

at points $(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*})=(\tilde{L};{\tilde{f}}_j,{\tilde{f}}_j^{*})\in$ $\tilde{\mathcal{L}}{(1|N)}^{*}\times {\left(L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\times L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\right)}^{*},$ $j\in \mathbb{N},$ respectively.

To show that the combined system is really a Hamiltonian system with respect to some other Poisson structure

$$\begin{array}{cc}\hfill \tilde{\mathsf{\Sigma }}& :\tilde{\mathcal{L}}{(1|N)}^{*}\times {\left(L_2({\mathbb{S}}^{1|N};\mathsf{\Lambda })\times L_2({\mathbb{S}}^{1|N};\mathsf{\Lambda })\right)}^{*}\to \hfill \\ \hfill & \to \tilde{\mathcal{L}}(1|N)\times \left(L_2({\mathbb{S}}^{1|N};\mathsf{\Lambda })\times L_2({\mathbb{S}}^{1|N};\mathsf{\Lambda })\right),\hfill \end{array}$$

(116)

we will consider preliminarily another extended Hamiltonian system on the combined phase space with respect to the canonical tensor Poisson structure (115) at some element $(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*})\in \tilde{\mathcal{L}}{(1|N)}^{*}\times {\left(L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\times L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\right)}^{*}$ generated, respectively, by the chosen smooth Hamiltonian functions ${\gamma }_j:\tilde{\mathcal{L}}{(1|N)}^{*}\times {\left(L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\times L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\right)}^{*}\to \mathbb{C},j\in \mathbb{N}$:

$$\begin{array}{cc}\hfill \mathsf{\partial }\widehat{L}\mathsf{\partial }{t}_j& =[P_{+}{\mathrm{grad}}_{\widehat{L}}{\widehat{\gamma }}_j(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*}),\widehat{L}-\mathsf{\partial }/\mathsf{\partial }y]-P_{+}[\mathrm{grad}{\widehat{\gamma }}_j(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*}),\widehat{L}-\mathsf{\partial }/\mathsf{\partial }y],\hfill \\ \hfill & \hfill \\ \hfill \mathsf{\partial }{\widehat{f}}_j/\mathsf{\partial }{t}_j& ={\mathrm{grad}}_{{\widehat{f}}^{*}}{\widehat{\gamma }}_j(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*}),\mathsf{\partial }{\widehat{f}}_j^{*}/\mathsf{\partial }{t}_j=-{\mathrm{grad}}_{{\widehat{f}}_j}{\widehat{\gamma }}_j(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*}),\hfill \end{array}$$

(117)

satisfying the following additional condition: there exists some smooth Backlund-type mapping $B:\tilde{\mathcal{L}}{(1|N)}^{*}\times {\left(L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\times L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\right)}^{*}\to \tilde{\mathcal{L}}{(1|N)}^{*}\times {\left(L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\times L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\right)}^{*}$ subject to which the following conditions

$$(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*})\stackrel{B}{\to }(\tilde{L}=\tilde{L}(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*});{\tilde{f}}_j={\widehat{f}}_j,{\tilde{f}}_j^{*}={\widehat{f}}_j)$$

(118)

hold for each $j\in \mathbb{N}$. To find the transformation (118), let us analyze the Hamiltonian functions variations $\delta {\widehat{\gamma }}_j(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*}){)|}_{\delta \widehat{L}=0}$, taking into account that ${\widehat{\gamma }}_j(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*})={\gamma }_j(\tilde{L}),j\in \mathbb{N},$ as well as the systems (112) and (117):

$$\begin{array}{cc}\hfill \delta \gamma (\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*}){|}_{\delta \widehat{L}=0}& =({\mathrm{grad}}_{\widehat{f}}{\widehat{\gamma }}_j|\delta {\widehat{f}}_j)+({\mathrm{grad}}_{{\widehat{f}}^{*}}{\widehat{\gamma }}_j|\delta {\widehat{f}}_j^{*})=\hfill \\ \hfill & =-(\mathsf{\partial }{\widehat{f}}_j^{*}/\mathsf{\partial }{t}_j|\delta {\widehat{f}}_j)+(\mathsf{\partial }{\widehat{f}}_j/\mathsf{\partial }{t}_j|\delta {\widehat{f}}_j^{*})=\hfill \\ \hfill & =-(\mathsf{\partial }{\tilde{f}}_j^{*}/\mathsf{\partial }{t}_j|\delta {\tilde{f}}_j)+(\mathsf{\partial }{\tilde{f}}_j^{*}/\mathsf{\partial }{t}_j|\delta {\tilde{f}}_j^{*})=\hfill \\ \hfill & =({\mathrm{grad}}_{\tilde{L}}^{*}{\gamma }_j{(\tilde{L})}_{+}{\tilde{f}}_j^{*}|\delta {\tilde{f}}_j)+({\mathrm{grad}}_{\tilde{L}}{\gamma }_j{(\tilde{L})}_{+}{\tilde{f}}_j|\delta {\tilde{f}}_j^{*})=\hfill \\ \hfill & =\mathrm{Tr}({\widehat{f}}_j^{*}{\mathrm{grad}}_{\tilde{L}}{\gamma }_j(\tilde{L})\delta {\widehat{f}}_j{D}_{\theta }{\mathsf{\partial }}^{-1})+\mathrm{Tr}({\mathrm{grad}}_{\tilde{L}}{\gamma }_j(\tilde{L})D_{\theta }{\mathsf{\partial }}_j^{-1}\widehat{f}\delta {\widehat{f}}_j^{*})=\hfill \\ \hfill & =\mathrm{Tr}\left({\mathrm{grad}}_{\tilde{L}}{\gamma }_j(\tilde{L})\delta \left({\widehat{f}}_j{D}_{\theta }{\mathsf{\partial }}^{-1}{\widehat{f}}_j^{*}\right)\right)=\mathrm{Tr}\left({\mathrm{grad}}_{\tilde{L}}{\gamma }_j(\tilde{L})\delta \tilde{L}\right){|}_{\delta \tilde{L}=0},\hfill \end{array}$$

(119)

where one easily obtains that $\delta \tilde{L}{|}_{{|}_{\delta \tilde{L}=0}}=\delta \left({\widehat{f}}_j{D}_{\theta }{\mathsf{\partial }}^{-1}{\widehat{f}}_j^{*}\right),$ or $\tilde{L}=\widehat{L}+{\widehat{f}}_j{D}_{\theta }{\mathsf{\partial }}^{-1}{\widehat{f}}_j^{*},j\in \mathbb{N}$. Thus, the resulting Backlund-type mapping (118) equals

$$(\widehat{L};{\widehat{f}}_j,{\widehat{f}}_j^{*})\stackrel{B}{\to }(\tilde{L}=\widehat{L}+{\widehat{f}}_j{D}_{\theta }{\mathsf{\partial }}^{-1}{\widehat{f}}_j^{*};{\tilde{f}}_j={\widehat{f}}_j,{\tilde{f}}_j^{*}={\widehat{f}}_j^{*})$$

(120)

for all $j\in \mathbb{N}$. The systems (117) are Hamiltonian with respect to the canonical tensor Poisson structure (115), and transform, owing to the above-constructed Backlund-type mapping (120), into the Poisson structure (116), which is calculated [42,63,64] via the following operator expression:

$$\tilde{\mathsf{\Sigma }}=B^{\prime }\left(\widehat{\mathsf{\Omega }}\otimes \widehat{J}\right)B^{\prime ,*},$$

(121)

giving rise to $j\in \mathbb{N}$ for the Poisson structure

$$\tilde{\mathsf{\Sigma }}=\left(\begin{array}{ccc}[{(...)}_{+},\tilde{L}-\mathsf{\partial }/\mathsf{\partial }y]-{[(...),\tilde{L}-\mathsf{\partial }/\mathsf{\partial }y]}_{+}+& -\tilde{f}{D}_{\theta }{\mathsf{\partial }}^{-1}\circ (...)& (...)\circ D_{\theta }{\mathsf{\partial }}^{-1}{\tilde{f}}^{*}\\ {(...)}_{+}\tilde{f}& 0& 1\\ -{(...)}_{+}^{*}{\tilde{f}}^{*}& -1& 0\end{array}\right)$$

on the phase space $\tilde{\mathcal{L}}{(1|N)}^{*}\times {\left(L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\times L_2({\mathbb{S}}^{1|N};{\mathsf{\Lambda }}_0)\right)}^{*}$, If one now chooses the Hamiltonian function as a Casimir one ${\gamma }_j\in I(\tilde{\mathcal{L}}{(1|N)}^{*}),j\in \mathbb{N},$ the resulting superflow

$$\mathsf{\partial }(\tilde{L};\tilde{f},{\tilde{f}}^{*}{)}^{⊺}/\mathsf{\partial }{t}_j:=\tilde{\mathsf{\Sigma }}\mathrm{grad}{\gamma }_j(\tilde{L};\tilde{f},{\tilde{f}}^{*})$$

(122)

proves to coincide exactly with the superflow (112). The results described above, in particular, the reasonings concerning the structure of the Backlund-type transformation on the extended adjoint superspace $\tilde{\mathcal{L}}{(1|N)}^{*}$, prove to be useful for constructing factorized superflows on the super-operator factors of the super-pseudo-differential operator element (107), generating the corresponding integrable superflows.

6. The Factorized Semi-Supersymmetric Hamiltonian Flows and Their Integrability

Return now to Problem 1 posed above, and make use of the results described above concerning the Hamiltonian superflows (109) and (110). Based on the reasonings above concerning the structure of the Backlund-type transformation on the extended adjoint superspace $\tilde{\mathcal{L}}{(1|N)}^{*}$, we can consider the expressions (107) and (111) that are related to each other as the corresponding Backlund-type invertible mapping

$$\begin{array}{c}exp\left(\tilde{\mathcal{L}}(1|N)\right)\times exp\left(\tilde{\mathcal{L}}(1|N)\right)\ni ({\tilde{F}}_n,{\tilde{Q}}_{n+p})\stackrel{T}{\to }\\ \\ \to (\tilde{L}={\tilde{F}}_n^{,-1}{\tilde{Q}}_{n+p},{\tilde{L}}^{\prime }={\tilde{Q}}_{n+p}{\tilde{F}}_n^{,-1})\in \tilde{\mathcal{L}}{(1|N)}^{*}\times \tilde{\mathcal{L}}{(1|N)}^{*}\end{array}$$

(123)

with the already-known canonical tensor super-Poisson operator

$$\tilde{\mathsf{\Psi }}:=\tilde{\mathsf{\Omega }}\otimes \tilde{\mathsf{\Omega }}:T^{*}(\tilde{\mathcal{L}}{(1|N)}^{*}\times \tilde{\mathcal{L}}{(1|N)}^{*})\to T(\tilde{\mathcal{L}}{(1|N)}^{*}\times \tilde{\mathcal{L}}{(1|N)}^{*})$$

(124)

on the phase superspace $\tilde{\mathcal{L}}{(1|N)}^{*}\times \tilde{\mathcal{L}}{(1|N)}^{*}$:

$$\tilde{\mathsf{\Psi }}:=\left(\begin{array}{cc}[{(...)}_{+},\tilde{L}-\mathsf{\partial }/\mathsf{\partial }y]-[(...),\tilde{L}-\mathsf{\partial }/\mathsf{\partial }y]& 0\\ 0& [{(...)}_{+},{\tilde{L}}^{\prime }-\mathsf{\partial }/\mathsf{\partial }y]-[(...),{\tilde{L}}^{\prime }-\mathsf{\partial }/\mathsf{\partial }y]\end{array}\right).$$

(125)

Taking into account the above-constructed Backlund-type transformation (123), one can find the related super-Poisson operator

$$\widehat{\mathsf{\Psi }}:T^{*}(\tilde{\mathcal{L}}{(1|N)}^{*}\times \tilde{\mathcal{L}}{(1|N)}^{*})\to T(\tilde{\mathcal{L}}{(1|N)}^{*}\times \tilde{\mathcal{L}}{(1|N)}^{*}),$$

(126)

acting already on the super-operator factors $({\tilde{F}}_n,{\tilde{Q}}_{n+p})\in exp\left(\tilde{\mathcal{L}}(1|N)\right)\times exp\left(\tilde{\mathcal{L}}(1|N)\right)$, making use of the operator expression (125) and the relationship like (121):

$$\widehat{\mathsf{\Psi }}=T^{\prime ,-1}\tilde{\mathsf{\Psi }}{\left(T^{\prime ,*}\right)}^{-1}.$$

(127)

Whence, by means of elementary-enough calculations, one ensues the following super-Poisson structure:

$$\begin{array}{cc}\hfill \widehat{\mathsf{\Psi }}& =\left(\begin{array}{cc}-{(1-{\tilde{L}}^{\prime }(\circ ){\tilde{L}}^{-1})}^{-1}\left({\tilde{F}}_n(·\right){\tilde{L}}^{-1})& {(1-{\tilde{L}}^{\prime }(\circ ){\tilde{L}}^{-1})}^{-1}((·){\tilde{F}}_n{\tilde{L}}^{-1})\\ {\tilde{F}}_n(·)-{(1-{\tilde{L}}^{\prime }(\circ ){\tilde{L}}^{-1})}^{-1}({\tilde{F}}_n(·){\tilde{L}}^{-1})\tilde{L}& {(1-{\tilde{L}}^{\prime }(\circ ){\tilde{L}}^{-1})}^{-1}((·){\tilde{F}}_n{\tilde{L}}^{-1})L\end{array}\right)\times \hfill \\ \hfill \times & \left(\begin{array}{cc}[{(...)}_{+},\tilde{L}-\mathsf{\partial }/\mathsf{\partial }y]-[(...),\tilde{L}-\mathsf{\partial }/\mathsf{\partial }y]& 0\\ 0& [{(...)}_{+},{\tilde{L}}^{\prime }-\mathsf{\partial }/\mathsf{\partial }y]-[(...),{\tilde{L}}^{\prime }-\mathsf{\partial }/\mathsf{\partial }y]\end{array}\right)\times \hfill \\ \hfill & \times \left(\begin{array}{cc}-L^{-1}{(1-{\tilde{L}}^{-1}(\circ ){\tilde{L}}^{\prime })}^{-1}(·){\tilde{F}}_n& (·){\tilde{F}}_n-{(1-{\tilde{L}}^{-1}(\circ ){\tilde{L}}^{\prime })}^{-1}(·){\tilde{F}}_n\\ {\tilde{Q}}_{n+p}{(1-{\tilde{L}}^{-1}(\circ ){\tilde{L}}^{\prime })}^{-1}(·)& {\tilde{F}}_n{(1-{\tilde{L}}^{-1}(\circ ){\tilde{L}}^{\prime })}^{-1}(·)\end{array}\right)\hfill \end{array}$$

(128)

on the phase superspace $exp\left(\tilde{\mathcal{L}}(1|N)\right)\times exp\left(\tilde{\mathcal{L}}(1|N)\right)$ under the conditions $\tilde{L}={\tilde{F}}_n^{-1}{\tilde{Q}}_{n+p}$ and ${\tilde{L}}^{\prime }={\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}\in \tilde{\mathcal{L}}{(1|N)}^{*}$.

The above-obtained super-Poisson structure (128) on the phase superspace $exp\left(\tilde{\mathcal{L}}(1|N)\right)\times exp\left(\tilde{\mathcal{L}}\mathsf{\Omega }(1|N)\right)$ can be used for constructing evolution flows, generated by the corresponding Casimir functionals ${\gamma }_j\in I\left(\tilde{\mathcal{L}}{(1|N)}^{*}\right), j\in \mathbb{N}$:

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{t}_j}}{\left({\tilde{F}}_n,{\tilde{Q}}_{n+p}\right)}^{⊺}=\widehat{\mathsf{\Psi }}({\tilde{F}}_n,{\tilde{Q}}_{n+p})\mathrm{grad}{\gamma }_j({\tilde{F}}_n,{\tilde{Q}}_{n+p}),$$

(129)

which naturally give rise to the corresponding factorized evolution superflows

$$\begin{array}{cc}\hfill \mathsf{\partial }{\tilde{F}}_n/\mathsf{\partial }{t}_j& =\mathrm{grad}\gamma {({\tilde{L}}^{\prime })}_{+}{\tilde{F}}_n-{\tilde{F}}_n\mathrm{grad}{\gamma }_j{(\tilde{L})}_{+}+{\tilde{M}}_{j,+}{\tilde{F}}_n,\hfill \\ \hfill \mathsf{\partial }{\tilde{Q}}_{n+p}/\mathsf{\partial }{t}_j& =\mathrm{grad}\gamma {({\tilde{L}}^{\prime })}_{+}{\tilde{Q}}_{n+p}-{\tilde{Q}}_{n+p}\mathrm{grad}{\gamma }_j{(\tilde{L})}_{+}+{\tilde{M}}_{j,-}{\tilde{Q}}_{n+p}\hfill \end{array}$$

(130)

on the super-operator components $({\tilde{F}}_n,{\tilde{Q}}_{n+p})\in exp\left(\tilde{\mathcal{L}}(1|N)\right)\times exp\left(\tilde{\mathcal{L}}\mathsf{\Omega }(1|N)\right)$, where the super-operators ${\tilde{M}}_{j,\pm }\in \tilde{\mathcal{L}}(1|N)$ satisfy the following compatibility conditions:

$$\begin{array}{cc}\hfill {\tilde{M}}_{j,-}{\tilde{Q}}_{n+p}-{\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}{\tilde{M}}_{j,+}{\tilde{F}}_n& ={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}{\gamma }_j({\tilde{L}}^{\prime }){\tilde{F}}_n,\hfill \\ \hfill {\tilde{M}}_{j,+}{\tilde{Q}}_{n+p}-{\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}{\tilde{M}}_{j,+}{\tilde{F}}_n& ={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}{\gamma }_j({\tilde{L}}^{\prime }){\tilde{F}}_n-{\tilde{F}}_n{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}\gamma (\tilde{L}),\hfill \end{array}$$

(131)

and, additionally, we took into account the obvious operator gradient relationship

$$\mathrm{grad}\gamma ({\tilde{F}}_n,{\tilde{Q}}_{n+p})=\left(\begin{array}{cc}-{\tilde{F}}_n^{-1}{\tilde{Q}}_{n+p}(·){\tilde{F}}_n^{-1}& 0\\ 0& (·){\tilde{F}}_n^{-1}\end{array}\right)\mathrm{grad}\gamma (\tilde{L}),$$

(132)

and

$$\mathrm{grad}\gamma ({\tilde{F}}_n,{\tilde{Q}}_{n+p})=\left(\begin{array}{cc}-{\tilde{F}}_n^{-1}(·){\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}& 0\\ 0& {\tilde{F}}_n^{-1}(·)\end{array}\right)\mathrm{grad}\gamma ({\tilde{L}}^{\prime }),$$

(133)

which is satisfied for arbitrary smooth functional $\gamma :\tilde{\mathcal{L}}{(1|N)}^{*}\to \mathbb{C}$ at points $\tilde{L}={\tilde{F}}_n^{-1}{\tilde{Q}}_{n+p}$ and ${\tilde{L}}^{\prime }={\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}\in \tilde{\mathcal{L}}{(1|N)}^{*}$, respectively. Thus, recalling Problem 1, posed before, of constructing factorized superflows on the super-operator components of generating orbit elements (107) and (111), we state that the systems (130) and (131) present its complete analytical solution. Namely, the following factorization theorem, generalizing the results in the works [32,35,61,62], holds.

Theorem 1. 

The operator superflow

$$\begin{array}{cc}\hfill \mathsf{\partial }{\tilde{F}}_n/\mathsf{\partial }t& =\mathrm{grad}\gamma {({\tilde{L}}^{\prime })}_{+}{\tilde{F}}_n-{\tilde{F}}_n\mathrm{grad}\gamma {(\tilde{L})}_{+}+{\tilde{M}}_{+}{\tilde{F}}_n,\hfill \\ \hfill \mathsf{\partial }{\tilde{Q}}_{n+p}/\mathsf{\partial }t& =\mathrm{grad}\gamma {({\tilde{L}}^{\prime })}_{+}{\tilde{Q}}_{n+p}-{\tilde{Q}}_{n+p}\mathrm{grad}\gamma {(L)}_{+}+{\tilde{M}}_{-}{\tilde{Q}}_{n+p}\hfill \end{array}$$

(134)

on the phase superspace $exp\left(\tilde{\mathcal{L}}(1|N)\right)\times exp\left(\tilde{\mathcal{L}}\mathsf{\Omega }(1|N)\right)$, generated by a Casimir invariant functional γ $\in I(\tilde{\mathcal{L}}{(1|N)}^{*})$, where the operators ${\tilde{M}}_{\pm }\in \tilde{\mathcal{L}}(1|N)$ satisfy the compatibility conditions

$$\begin{array}{cc}\hfill {\tilde{M}}_{-}{\tilde{Q}}_{n+p}-{\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}{\tilde{M}}_{+}{\tilde{F}}_n& ={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}\gamma ({\tilde{L}}^{\prime }){\tilde{F}}_n,\hfill \\ \hfill {\tilde{M}}_{+}{\tilde{Q}}_{n+p}-{\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}{\tilde{M}}_{+}{\tilde{F}}_n& ={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}\gamma ({\tilde{L}}^{\prime }){\tilde{F}}_n-{\tilde{F}}_n{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\mathrm{grad}\gamma (\tilde{L}),\hfill \end{array}$$

(135)

is Hamiltonian with respect to the super-Poisson structure (128) and factorizes the Lax-type flows

$$\mathsf{\partial }\tilde{L}/\mathsf{\partial }t=[\mathrm{grad}\gamma {(\tilde{L})}_{+},\tilde{L}]+\mathsf{\partial }/\mathsf{\partial }y\mathrm{grad}\gamma {(\tilde{L})}_{+}$$

(136)

on the seed element $\tilde{L}={\tilde{F}}_n^{-1}{\tilde{Q}}_{n+p}\in \tilde{\mathcal{L}}(1|N)$ and

$$\mathsf{\partial }{\tilde{L}}^{\prime }/\mathsf{\partial }t=[\mathrm{grad}\gamma {({\tilde{L}}^{\prime })}_{+},{\tilde{L}}^{\prime }]+\mathsf{\partial }/\mathsf{\partial }y\mathrm{grad}\gamma {({\tilde{L}}^{\prime })}_{+}$$

(137)

with respect to the operator elements $\tilde{L}={\tilde{F}}_n^{-1}{\tilde{Q}}_{n+p}$ $\in \tilde{\mathcal{L}}{(1|N)}^{*}$ and ${\tilde{L}}^{\prime }={\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}\in \tilde{\mathcal{L}}{(1|N)}^{*}$, respectively.

As a simple consequence from Theorem 1, one derives the following proposition.

Proposition 3. 

There exist such smooth mappings $\tilde{T},\tilde{S}:$ $\mathbb{R}$ $\to exp\left(\tilde{\mathcal{L}}{(1|N)}_{-}\right)$ to the formal super-pseudo-differential operator subgroup $exp\left(\tilde{\mathcal{L}}{(1|N)}_{\_}\right)$, satisfying the linear evolution superflows

$$\mathsf{\partial }\tilde{S}/\mathsf{\partial }t-\mathit{grad}\gamma {({\tilde{L}}^{\prime })}_{+}\tilde{S}=0,\mathsf{\partial }\tilde{T}/\mathsf{\partial }t-\mathit{grad}\gamma {(\tilde{L})}_{+}\tilde{T}=0$$

under the Cauchy data $\tilde{T}{{|}_{t=0}=\overline{A},\tilde{S}|}_{t=0}=\overline{B}\in exp\left(\mathcal{L}(1|N)\_\right)$, generated, respectively, by the Lie superalgebra elements $grad\gamma {(\tilde{L})}_{+}$ and $grad\gamma {({\tilde{L}}^{\prime })}_{+}\in \tilde{\mathcal{L}}{(1|N)}_{+}$, that

$${\tilde{\mathrm{F}}}_n:=\tilde{S}{\overline{\mathrm{F}}}_n{\tilde{T}}^{-1},{\tilde{\mathrm{Q}}}_{n+p}^{(l)}:=\tilde{S}{\overline{\mathrm{Q}}}_{n+p}^{(l)}{\tilde{T}}^{-1},$$

(138)

where, by definition, ${\tilde{\mathrm{F}}}_n:={\tilde{F}}_n,$ ${\tilde{\mathrm{Q}}}_{n+p}:={\tilde{Q}}_{n+p}-{\tilde{F}}_n\mathsf{\partial }/\mathsf{\partial }y$ and the operator elements ${\tilde{\mathrm{F}}}_n{|}_{t=0}={\overline{F}}_n$ and ${\tilde{\mathrm{Q}}}_{n+p}{|}_{t=0}={\overline{\mathrm{Q}}}_{n+p}:={\overline{Q}}_{n+p}-{\overline{F}}_n\mathsf{\partial }/\mathsf{\partial }y\in exp\left(\mathcal{L}{(1|N)}_{+}\right)$ are taken to be constant with respect to the evolution parameter $t\in \mathbb{R}$.

Proof. 

It is enough to check, using (135), that the group elements (138) really satisfy the factorized evolution equation (134). Nowm based on Proposition 3, we can take into account, with no loss of generality, that the group elements $\tilde{S},\tilde{T}\in exp\left(\tilde{\mathcal{L}}{(1|N)}_{-}\right)$ for all $t\in \mathbb{R}$ can be represented as the operator series

$$\tilde{S}(x,\theta ;y,t)\sim I+\sum _{j\in {\mathbb{Z}}_{+}}{b}_j(x,\theta ;y,t){\mathsf{\partial }}^{-j},\tilde{T}(x,\theta ;y,t)\sim I+\sum _{j\in {\mathbb{Z}}_{+}}{a}_j(x,\theta ;y,t){\mathsf{\partial }}^{-j},$$

(139)

whose coefficients can be found recurrently from the expressions (138), rewritten in the following form useful for calculations:

$$\begin{array}{c}\tilde{S}(x,\theta ;y,t)\circ {\overline{F}}_n={\tilde{F}}_n\circ \tilde{T}(x,\theta ;y,t),\\ \tilde{S}(x,\theta ;y,t)\circ ({\overline{Q}}_{n+p}-{\overline{F}}_n\mathsf{\partial }/\mathsf{\partial }y)=({\tilde{Q}}_{n+p}-{\tilde{F}}_n\mathsf{\partial }/\mathsf{\partial }y)\circ \tilde{T}(x,\theta ;y,t),\end{array}$$

(140)

where the supergroup elements ${\tilde{F}}_n{|}_{t=0}={\overline{F}}_n$ and ${\tilde{Q}}_{n+p}{|}_{t=0}={\overline{Q}}_{n+p}\in exp\left(\tilde{\mathcal{L}}{(1|N)}_{+}\right)$ are taken a priori constant both in $t\in \mathbb{R}$ and $x\in {\mathbb{S}}^1$, motivated both by the compatible evolution superflows

$$\begin{array}{cc}\hfill \mathsf{\partial }{\tilde{\mathrm{F}}}_n/\mathsf{\partial }t& =\mathrm{grad}\gamma {({\tilde{L}}^{\prime })}_{+}{\tilde{\mathrm{F}}}_n-{\tilde{\mathrm{F}}}_n\mathrm{grad}\gamma {(\tilde{L})}_{+},\hfill \\ \hfill \mathsf{\partial }{\tilde{\mathrm{Q}}}_{n+p}/\mathsf{\partial }t& =\mathrm{grad}\gamma {({\tilde{L}}^{\prime })}_{+}{\tilde{\mathrm{Q}}}_{n+p}-{\tilde{\mathrm{Q}}}_{n+p}\mathrm{grad}\gamma {(\tilde{L})}_{+},\hfill \end{array}$$

(141)

giving rise to the evolution superflows (109) and (110), as well as to the operator relationships

$$\begin{array}{cc}\hfill ({\tilde{F}}_n^{-1}{\tilde{Q}}_{n+p}-\mathsf{\partial }/\mathsf{\partial }y)\tilde{T}(x,\theta ;y,t)& =\tilde{T}(x,\theta ;y,t)({\overline{F}}_n^{-1}{\overline{Q}}_{n+p}-\mathsf{\partial }/\mathsf{\partial }y),\hfill \\ \hfill & \hfill \\ \hfill ({\tilde{Q}}_{n+p}{\tilde{F}}_n^{-1}-\mathsf{\partial }/\mathsf{\partial }y)\tilde{S}(x,\theta ;y,t)& =\tilde{S}(x,\theta ;y,t)({\overline{Q}}_{n+p}{\overline{F}}_n^{-1}-\mathsf{\partial }/\mathsf{\partial }y),\hfill \end{array}$$

(142)

determining the related Casimir invariants and following directly from (140). □

The results obtained above demonstrate, in particular, that the relationships (134) and (135) can be algorithmically used for constructing new many-component super-integrable Hamiltonian systems, generated by a suitably chosen Casimir functional $\gamma \in$ $I(exp\left(\mathcal{L}{(1|N)}^{*}\right))$. This and other related aspects of this both interesting and important problem of classifying such super-Hamiltonian systems are planned to be analyzed in detail in a work under preparation.

7. Conclusions

In this work, we first described the basic preliminaries of differential-geometric relationships on the calculus of the supercircle ${\mathbb{S}}^{1|N}$ and presented a derivation of superconformal affine Lie superalgebras $\widehat{\mathcal{K}}(1|N),N\in \mathbb{N}$. Their central extensions for $N=1,2,3$ allowed us to construct infinite hierarchies of semi-supersymmetric integrable Hamiltonian flows on related functional supermanifolds as coadjoint orbits of these superconformal affine Lie superalgebras $\widehat{\mathcal{K}}(1|N=1,2,3)$ generated by the corresponding Casimir invariants. The Lie-algebraic analysis of these coadjoint orbits made it possible to state the bi-Hamiltonicity of these superflows and build related super-Poisson structures on the functional supermanifolds. In addition, we also generalized these results subject to the suitably factorized super-pseudo-differential Lax-type linear problems, taking into account the devised-before algebro-analytic constructions, devoted to Lie-superalgebraic properties of factorized Lax-type representations and the factorized Hamiltonian systems, respectively. We also succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of centrally extended pseudo-differential operator superalgerbras.