BiHom–Lie Brackets and the Toda Equation

Abstract

We introduce a BiHom-type skew-symmetric bracket on general linear Lie algebra $GL\left(V\right)$ built from two commuting inner automorphisms $\alpha =A{d}_{\psi }$ and $\beta =A{d}_{\varphi }$, with $\psi ,\varphi \in GL\left(V\right)$ and integers $i,j$. We prove that $(GL\left(V\right),{[·,·]}_{(\psi ,\varphi )}^{(i,j)},\alpha ,\beta )$ is a BiHom–Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $\varphi =\psi$ with $(i,j)=(0,0)$, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via $\tilde{L}={\psi }^{-1}L\psi$, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times 2$ blocks, we derive explicit trigonometric and hyperbolic formulae that make symmetry constraints (e.g., tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times 2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values.

1. Introduction

As one of the earliest examples of a nonlinear completely integrable system, the Toda lattice hierarchy has attracted sustained interest from both mathematicians and physicists. After the lattice was shown to admit solutions expressible via elliptic functions, it was soon observed to possess multi-soliton solutions with elastic collisions. This parallel with the KdV equation, together with the subsequent Lax-pair formulation, led to its recognition as an integrable system. Many mathematical treatments and generalizations of the original structures continue. In particular, one common thread is the relation of infinite-dimensional Hamiltonian and Lagrangian dynamical systems to infinite-dimensional Lie algebras [1,2,3]. This algebraic viewpoint, where invariance, equivariance, and symmetry constraints organize the dynamics, is also the focus of the present paper.

A Lie bracket is a basic instrument for encoding the algebraic content behind nonlinear dynamics. Therefore, deforming the bracket is a natural way to organize non-standard evolutions and to track symmetry versus symmetry breaking. In the context of conformal field theory (CFT) and quantum algebra [4,5,6,7,8], Hom- [9] and BiHom-type [10] structures arise naturally, for instance in deformations of Virasoro-type algebras. Motivated by this, we employ a BiHom-type deformation that twists the commutator by two commuting inner automorphisms; see [10,11,12,13] for background and [4,5,6,7,8,14,15,16] for related appearances in quantum algebra and high-energy physics.

In light of this, we present a deformed (non-standard) version of the finite nonperiodic Toda lattice obtained by replacing the commutator in the Lax equation with a BiHom-type bracket produced by the Makhlouf–Silvestrov construction. Concretely, we work with two commuting one-parameter families

$$\psi ,\varphi :R_1⟶GL\left(V\right),$$

and set ${\alpha }_s:=A{d}_{\psi \left(s\right)}$, ${\beta }_s:=A{d}_{\varphi \left(s\right)}$. For two integers i and j, we determine the unique skew-symmetric bilinear form within a natural ansatz that makes the BiHom–Jacobi identity hold. This bracket generalizes, and for $i+j\ne 1$ also extends beyond, the Hom-type bilinear form considered in [13]. Beyond the abstract construction, we analyze three explicit deformation families (i.e., scalar dilations, planar rotations, and planar hyperbolic rotations) and compute the resulting $2\times 2$ block dynamics in detail, with particular attention to tracelessness, symmetry/equivariance, and (non-)isospectrality indicators. Compared with the classical and the Hom–Toda equations, the Bihom-type bracket could give more generalized solutions in the case of “Scalar dilations” (see Section 4.1), the case of asymmetry of “Planar rotations ” (see Section 4.2), and the case of asymmetry of “Planar hyperbolic rotations” (see Section 4.3), which cannot be realized by the classical or Hom-type brackets.

To further extend the scope, following the method of Li and He [17], we formulate the weakly coupled Toda lattice with an indefinite metric and its non-standard counterpart. In the two-dimensional case, we combine a direct Lax computation with the inverse-scattering scheme in [18] to obtain unified explicit solutions and to clarify and correct certain formulas in [19].

The layout of the paper is as follows. In Section 2, we fix the notations, recall the BiHom–Jacobi identity, and define the deformed bracket. We also establish the uniqueness of the skew-symmetric operator within the bilinear ansatz. In Section 3, we describe, in elementary group-theoretic terms, three concrete one-parameter deformations that will be used later. In Section 4, we specialize the deformed Lax equation to three families (scalar, rotation, and hyperbolic rotation), extract their structural consequences, and provide closed $2\times 2$ formulas that make the underlying symmetry constraints explicit. Section 5 gives a Miura-type relation clarifying when the deformed flow is conjugate to a Toda-type Lax equation and when asymmetry induces symmetry breaking. Section 6 treats the weakly coupled case with indefinite metrics, deriving explicit solutions in the two-dimensional normal form and explaining the corrections mentioned above.

In this paper, integrability is understood in the classical Lax-pair sense: a matrix evolution equation is called integrable if there exists a matrix-valued functional $M\left(L\right)$ such that

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}L=[M\left(L\right),L].$$

This structure guarantees that the flow is isospectral, i.e., the eigenvalues of $L\left(t\right)$ remain constant in time. Conversely, if ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}L$ cannot be written as a commutator, then the spectrum of $L\left(t\right)$ is not preserved, and the system fails to be integrable in the Lax sense.

2. Deformations of the Lie Bracket

In this section, we define the deformed (BiHom) Lie bracket, which will be used throughout, and then, we also give the corresponding Lax equation for the Toda lattice.

Let $V={\mathbb{R}}^n$, and let $GL\left(V\right)$ be the space of all real $n\times n$ matrices. For $L\in GL\left(V\right)$, we denote by $L_{>0}$ (resp. $L_{<0}$) the strictly upper (resp. strictly lower) triangular part of L, and set

$$B\left(L\right):=L_{>0}-L_{<0}.$$

Note that $B\left(L\right)$ is skew-symmetric with respect to the upper/lower triangular splitting in the sense that $B{\left(L\right)}^{\top }=-B\left(L\right)$ whenever L is symmetric. The skew-symmetry will be mirrored by the BiHom-type deformation below.

We consider two $C^1$ (continuously differentiable) one-parameter maps

$$\psi ,\varphi :R_1⟶GL\left(V\right), s\mapsto \psi \left(s\right), \varphi \left(s\right),$$

where $R_1\subseteq \mathbb{R}$ is an interval, such that the matrices $\psi \left(s\right)$ and $\varphi \left(s\right)$ commute for each $s\in R_1$. We write ${\alpha }_s:={Ad}_{\psi \left(s\right)}$ and ${\beta }_s:={Ad}_{\varphi \left(s\right)}$ for their adjoint actions on $GL\left(V\right)$. For integer exponents, we use the convention $\psi {\left(s\right)}^{-1}=\psi {\left(s\right)}^{ (-1)}$, where the former denotes the inverse of $\psi \left(s\right)$, and the latter denotes the exponent -1. The same convention applies to $\varphi \left(s\right)$.

For completeness, we recall the Hom–Jacobi identity ([11,12]) in terms of a single twisting map $\alpha$:

$$\left[\alpha \right(a),[b,c\left]\right]+\left[\alpha \right(b),[c,a\left]\right]+\left[\alpha \right(c),[a,b\left]\right]=0.$$

(1)

Further, a BiHom–Lie algebra ([10]) over a field k is a quadruple $(L,[ , ],\alpha ,\beta )$, where L is a k-linear space, $\alpha :L\to L$, $\beta :L\to L$, $[ , ]:L\otimes L\to L$ are linear maps, with the notation $[ , ](a\otimes b)=[a,b]$, satisfying the following conditions, for all $a,b,c\in L$:

$$\alpha \circ \beta =\beta \circ \alpha ,$$

(2)

$$\left[\beta \right(a),\alpha (b\left)\right]=-\left[\beta \right(b),\alpha (a\left)\right],$$

(3)

$$\alpha \left(\right[a,b\left]\right)=\left[\alpha \right(a),\alpha (b\left)\right], \beta \left(\right[a,b\left]\right)=\left[\beta \right(a),\beta (b\left)\right],$$

(4)

$$[{\beta }^2\left(a\right),[\beta \left(b\right),\alpha \left(c\right)]]+[{\beta }^2\left(b\right),[\beta \left(c\right),\alpha \left(a\right)]]+[{\beta }^2\left(c\right),[\beta \left(a\right),\alpha \left(b\right)]]=0.$$

(5)

In what follows, we construct an explicit bracket ${[·,·]}_{(\psi ,\varphi )}^{(i,j)}$, which satisfies (5), with respect to $({\alpha }_s,{\beta }_s)$, and exhibits the required BiHom skew-symmetry between the $\psi$-actions and $\varphi$-actions (see Appendix A for a uniqueness statement within a bilinear ansatz).

Fix $s\in R_1$ and integers $i,j\in \mathbb{Z}$. For $A,B\in GL\left(V\right)$, set

$${[A,B]}_{(\psi ,\varphi )}^{(i,j)}\left(s\right):=\psi \left(s\right) A \psi {\left(s\right)}^{ i}\varphi {\left(s\right)}^{ j} B \varphi {\left(s\right)}^{-1} - \varphi \left(s\right) B \psi {\left(s\right)}^{ i+2}\varphi {\left(s\right)}^{ j-2} A \psi {\left(s\right)}^{-1},$$

(6)

which is the form used in the uniqueness derivation in Appendix A. We always suppress the parameter s from the notation without confusion. Based on this, we state the following theorem.

Theorem 1.

Let $\psi \left(s\right),\varphi \left(s\right)\in GL\left(V\right)$ commute for each $s\in R_1$ and $i,j\in \mathbb{Z}$. Then, the quadruple

$$\left(GL\left(V\right),{[·,·]}_{(\psi ,\varphi )}^{(i,j)}\left(s\right),{\alpha }_s,{\beta }_s\right)$$

is a BiHom–Lie algebra.

Proof. 

Since the matrices $\psi \left(s\right)$ and $\varphi \left(s\right)$ commute, for each $A\in GL\left(V\right)$, we have the following:

$$\begin{array}{cc}\hfill {\alpha }_s\circ {\beta }_s\left(A\right)& ={Ad}_{\psi \left(s\right)}\circ {Ad}_{\varphi \left(s\right)}\left(A\right)\hfill \\ \hfill & =\psi \left(s\right)\varphi \left(s\right)A\varphi {\left(s\right)}^{-1}\psi {\left(s\right)}^{-1}\hfill \\ \hfill & =\varphi \left(s\right)\psi \left(s\right)A\psi {\left(s\right)}^{-1}\varphi {\left(s\right)}^{-1}\hfill \\ \hfill & ={Ad}_{\varphi \left(s\right)}\circ {Ad}_{\psi \left(s\right)}\left(A\right)\hfill \\ \hfill & ={\beta }_s\circ {\alpha }_s\left(A\right).\hfill \end{array}$$

Thus, we conclude ${\alpha }_s\circ {\beta }_s={\beta }_s\circ {\alpha }_s$, which satisfies (2). From this, we can obtain the following for any $A,B\in GL\left(V\right)$,

$$\begin{array}{cc}\hfill {\alpha }_s\left({[A,B]}_{(\psi ,\varphi )}^{(i,j)}\left(s\right)\right)& ={Ad}_{\psi \left(s\right)}(\psi \left(s\right) A \psi {\left(s\right)}^{ i}\varphi {\left(s\right)}^{ j} B \varphi {\left(s\right)}^{-1} - \varphi \left(s\right) B \psi {\left(s\right)}^{ i+2}\varphi {\left(s\right)}^{ j-2} A \psi {\left(s\right)}^{-1})\hfill \\ \hfill & =\psi \left(s\right)(\psi \left(s\right) A \psi {\left(s\right)}^{ i}\varphi {\left(s\right)}^{ j} B \varphi {\left(s\right)}^{-1} - \varphi \left(s\right) B \psi {\left(s\right)}^{ i+2}\varphi {\left(s\right)}^{ j-2} A \psi {\left(s\right)}^{-1})\psi {\left(s\right)}^{-1}\hfill \\ \hfill & =\psi \left(s\right) \left(\psi \left(s\right)A\psi {\left(s\right)}^{-1}\right) \psi {\left(s\right)}^{ i}\varphi {\left(s\right)}^{ j} \left(\psi \left(s\right)B\psi {\left(s\right)}^{-1}\right) \varphi {\left(s\right)}^{-1}\hfill \\ \hfill & - \varphi \left(s\right) \left(\psi \left(s\right)B\psi {\left(s\right)}^{-1}\right) \psi {\left(s\right)}^{ i+2}\varphi {\left(s\right)}^{ j-2} \left(\psi \left(s\right)A\psi {\left(s\right)}^{-1}\right) \psi {\left(s\right)}^{-1}\hfill \\ \hfill & ={[{Ad}_{\psi \left(s\right)}\left(A\right),{Ad}_{\psi \left(s\right)}\left(B\right)]}_{(\psi ,\varphi )}^{(i,j)}\left(s\right)\hfill \\ \hfill & ={[{\alpha }_s\left(A\right),{\alpha }_s\left(B\right)]}_{(\psi ,\varphi )}^{(i,j)}\left(s\right).\hfill \end{array}$$

Similarly, we get ${\beta }_s\left({[A,B]}_{(\psi ,\varphi )}^{(i,j)}\left(s\right)\right)={[{\beta }_s\left(A\right),{\beta }_s\left(B\right)]}_{(\psi ,\varphi )}^{(i,j)}\left(s\right)$. These results satisfy (4). Finally, we maintain the asymmetric property (3) and the BiHom–Jacobi identity (5) as shown in Appendix A. Therefore, we have proved the theorem. □

Within the bilinear ansatz considered in Appendix A, Formula (6) is characterized by the choice of i and j.

Remark 1

(Specializations).

• If $\psi \left(s\right)=\varphi \left(s\right)$, then

  $${[A,B]}_{(\psi ,\psi )}^{(i,j)}\left(s\right)=\psi \left(s\right)A \psi {\left(s\right)}^{ i+j}B \psi {\left(s\right)}^{-1}-\psi \left(s\right)B \psi {\left(s\right)}^{ i+j}A \psi {\left(s\right)}^{-1}.$$
  In particular, for $(i,j)=(-1,0)$, this simplifies to

  $${[A,B]}_{(\psi ,\psi )}^{(-1,0)}\left(s\right)=\psi \left(s\right)A\psi {\left(s\right)}^{-1} B \psi {\left(s\right)}^{-1}-\psi \left(s\right)B\psi {\left(s\right)}^{-1} A \psi {\left(s\right)}^{-1},$$

  which is not equal to $\psi \left(s\right)[A,B]\psi {\left(s\right)}^{-1}$ in general. This case has been discussed in [13].
  Specially, for $(i,j)=(0,0)$, this reduces to

  $${[A,B]}_{(\psi ,\psi )}^{(0,0)}\left(s\right)=\psi \left(s\right)AB \psi {\left(s\right)}^{-1}-\psi \left(s\right)BA \psi {\left(s\right)}^{-1},$$

  which is equal to $\psi \left(s\right)[A,B]\psi {\left(s\right)}^{-1}$. At the level of the Lax equation, however, the resulting flow is gauge-equivalent to a Toda-type equation with a conjugated triangular projection; see Section 5.
• If $\psi \left(s\right)=\varphi \left(s\right)={\mathrm{Id}}_V$, we recover the ordinary commutator.

Following from [20,21,22], the (finite nonperiodic) Toda lattice is written as the Lax equation

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}L=[B\left(L\right),L], B\left(L\right):=L_{>0}-L_{<0}.$$

(7)

Here, $B\left(L\right)$ realizes the classical antisymmetrization with respect to the upper/lower triangular decomposition.

Fix $s\in R_1$. Replacing the commutator in (7) by the BiHom-type bracket (6), we obtain the deformed (BiHom–Toda) Lax equation

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}L={[B\left(L\right),L]}_{(\psi ,\varphi )}^{(i,j)}\left(s\right),$$

(8)

that is,

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}L=\psi \left(s\right)B\left(L\right) \psi {\left(s\right)}^{ i}\varphi {\left(s\right)}^{ j}L \varphi {\left(s\right)}^{-1}-\varphi \left(s\right)L \psi {\left(s\right)}^{ i+2}\varphi {\left(s\right)}^{ j-2}B\left(L\right) \psi {\left(s\right)}^{-1}.$$

(9)

Here, s is a deformation parameter. Two important special cases are worth recording:

• If $\psi \left(s\right)=\varphi \left(s\right)$ and $(i,j)=(0,0)$, then the flow is gauge-equivalent to a Toda-type equation with a conjugated triangular projection (see Section 5 for the precise statement and proof).
• If $\psi \left(s\right)=\varphi \left(s\right)={\mathrm{Id}}_V$, (8) reduces to (7).

We will build on (8) in the subsequent sections.

3. One-Parameter Deformations via Conjugations

This section records the concrete one-parameter families of matrices, whose adjoint actions we will use to build the deformed bracket from Section 2 and the corresponding deformed Toda flows.

For a $2\times 2$ matrix M, we write

$$\mathcal{E}\left(M\right):=\mathrm{diag}(I_{n-2},M)\in GL\left({\mathbb{R}}^n\right),$$

(10)

so that conjugation by $\mathcal{E}\left(M\right)$ acts nontrivially only on the last $2\times 2$ block and trivially on the $(n-2)\times (n-2)$ identity block.

We shall use the following commuting one-parameter families $\psi ,\varphi :R_1\to GL\left({\mathbb{R}}^n\right)$.

• (Scalar dilations). For $r\in {\mathbb{R}}^{\times }$ and $p,q\in \mathbb{R}$, set

  $$\psi \left(r\right):=r^p{I}_n, \varphi \left(r\right):=r^q{I}_n.$$

  (11)
  Then, $\psi \left(r_1\right)\psi \left(r_2\right)=\psi \left(r_1{r}_2\right)$, $\psi {\left(r\right)}^{-1}=\psi \left(r^{-1}\right)$, and similarly for $\varphi$; moreover, ${\mathrm{Ad}}_{\psi \left(r\right)}={\mathrm{Id}}_{{\mathbb{R}}^n}$ on $GL\left({\mathbb{R}}^n\right)$. This family reflects a uniform scaling symmetry of the bracketed dynamics.
• (Planar rotations (Euclidean case)). For $\theta \in \mathbb{R}$, define

  $$R_2\left(\theta \right):=\left(\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right), \psi \left(\theta \right):=\mathcal{E} \left(R_2\left(\theta \right)\right), \varphi \left(\theta \right):=\mathcal{E} \left(R_2(\pm \theta )\right).$$

  (12)
  We have $R_2\left({\theta }_1\right)R_2\left({\theta }_2\right)=R_2({\theta }_1+{\theta }_2)$ and $R_2{\left(\theta \right)}^{-1}=R_2(-\theta )$; hence, $\psi \left({\theta }_1\right)\psi \left({\theta }_2\right)=\psi ({\theta }_1+{\theta }_2)$; $\psi \left(\theta \right)$ commutes with $\varphi \left(\theta \right)$ for either sign choice. We distinguish the symmetric choice $\varphi \left(\theta \right)=\psi \left(\theta \right)$ and the asymmetric choice $\varphi \left(\theta \right)=\mathcal{E}\left(R_2(-\theta )\right)$:

  -

  Symmetric choice $\varphi \left(\theta \right)=\psi \left(\theta \right)$. For $i,j\in \mathbb{Z}$, one has the identity

  $${[A,B]}_{(\psi ,\psi )}^{(i,j)}\left(\theta \right)=\psi \left(\theta \right) A \psi {\left(\theta \right)}^{ i+j} B \psi {\left(\theta \right)}^{-1}-\psi \left(\theta \right) B \psi {\left(\theta \right)}^{ i+j} A \psi {\left(\theta \right)}^{-1}.$$

  (13)

  In particular, for $(i,j)=(0,0)$, this reduces to

  $${[A,B]}_{(\psi ,\psi )}^{(0,0)}\left(\theta \right)=\psi \left(\theta \right)AB \psi {\left(\theta \right)}^{-1}-\psi \left(\theta \right)BA \psi {\left(\theta \right)}^{-1}.$$

  The associated deformed Lax flow is gauge-equivalent to a Toda-type equation with a conjugated triangular projection, see Theorem 2. It coincides with the classical Toda flow precisely when $\psi \left(\theta \right)$ preserves the upper/lower triangular splitting (e.g., when $\psi \left(\theta \right)$ is diagonal).

  -

  Asymmetric choice $\varphi \left(\theta \right)=\mathcal{E}\left(R_2(-\theta )\right)$. This produces a genuinely different bracket whose $2\times 2$-block effect will be analyzed in Section 4.

• (Planar hyperbolic rotations (Lorentzian case)). For $\lambda \in \mathbb{R}$, define

  $$H_2\left(\lambda \right):=\left(\begin{array}{cc}cosh\lambda & sinh\lambda \\ sinh\lambda & cosh\lambda \end{array}\right), \psi \left(\lambda \right):=\mathcal{E} \left(H_2\left(\lambda \right)\right), \varphi \left(\lambda \right):=\mathcal{E} \left(H_2(\pm \lambda )\right).$$

  (14)
  Then, $H_2\left({\lambda }_1\right)H_2\left({\lambda }_2\right)=H_2({\lambda }_1+{\lambda }_2)$ and $H_2{\left(\lambda \right)}^{-1}=H_2(-\lambda )$; so, the additivity and invertibility properties hold for $\psi ,\varphi$ as in (12). Again, $\psi \left(\lambda \right)$ commutes with $\varphi \left(\lambda \right)$ for either sign choice. We likewise distinguish the following:

  -

  Symmetric choice $\varphi \left(\lambda \right)=\psi \left(\lambda \right)$. In particular, for $(i,j)=(0,0)$, the deformed Lax flow is gauge-equivalent to a Toda-type equation with a conjugated triangular projection (see Theorem 2); it coincides with the classical Toda flow precisely when $\psi \left(\lambda \right)$ preserves the upper/lower triangular splitting (e.g., $\psi \left(\lambda \right)$ is diagonal).

  -

  Asymmetric choice $\varphi \left(\lambda \right)=\mathcal{E}\left(H_2(-\lambda )\right)$. This leads to non-conjugate deformations; see Section 4 for $2\times 2$ block formulas and trace properties.

Recalling the bracket from (6), the previous families induce the following specializations that will be used repeatedly:

(A^′)

Scalar dilations. If $\psi \left(r\right)=r^p{I}_n$ and $\varphi \left(r\right)=r^q{I}_n$, then

$${[A,B]}_{(\psi ,\varphi )}^{(i,j)}\left(r\right)=r^{ p(i+1)+q(j-1)} [A,B].$$

(15)

Consequently, in the deformed Lax equation $\dot{L}={[B\left(L\right),L]}_{(\psi ,\varphi )}^{(i,j)}\left(r\right)$, the right-hand side is a constant multiple of the classical one.

(B^′)

Rotations. With $\psi \left(\theta \right)=\mathcal{E}\left(R_2\left(\theta \right)\right)$ and either $\varphi \left(\theta \right)=\psi \left(\theta \right)$ or $\varphi \left(\theta \right)=\mathcal{E}\left(R_2(-\theta )\right)$, Formula (6) specializes to explicit trigonometric combinations. In the symmetric case with $(i,j)=(0,0)$, the associated Lax flow is gauge-equivalent to a Toda-type equation with conjugated projection (Theorem 2); see Section 4 for $2\times 2$ block formulas in the asymmetric case.

(C^′)

Hyperbolic rotations. The situation is analogous to Section 3. The identities for cosh and sinh guarantee that the required manipulations carry over verbatim. In the symmetric case with $(i,j)=(0,0)$, the flow is gauge-equivalent to a Toda-type equation with conjugated projection; the asymmetric choice provides non-conjugate deformations discussed in Section 4.

Remark 2.

In all three families, we have $\psi \left(0\right)=\varphi \left(0\right)=I_n$ (or $\psi \left(1\right)=\varphi \left(1\right)=I_n$ in the multiplicative scalar case (11)). Thus, the parameter s controls a deformation anchored at the identity, preserving the natural symmetry at zero deformation required by the setup in Section 2.

4. Behavior and Solutions in Three Canonical Deformations

We now specialize the deformed Lax Equation (8) to the three one-parameter families recorded in Section 3 and extract consequences for the dynamics. Throughout, $i,j\in \mathbb{Z}$, and $B\left(L\right)=L_{>0}-L_{<0}$. Lengthy $2\times 2$ block computations are deferred to Appendix B; here we state the structural facts that will be used later.

4.1. Case I: Scalar Dilations

In this case, the deformation only induces constant time rescaling of the classical Toda flow. All conserved quantities are preserved, and the phase portrait remains identical to the classical system—no changes to symmetry, integrability, or qualitative dynamics beyond adjusting the speed of evolution.

Let $\psi \left(r\right)=r^p{I}_n$, $\varphi \left(r\right)=r^q{I}_n$ with $r\in {\mathbb{R}}^{\times }$. By (6), one has

$${[B\left(L\right),L]}_{(\psi ,\varphi )}^{(i,j)}\left(t\right)=\alpha [B\left(L\right),L]\left(t\right), \alpha :=r^{ p(i+1)+q(j-1)}.$$

(16)

Proposition 1.

Let $L_{\mathrm{Toda}}\left(t\right)$ be the solution to the classical Toda flow $\dot{L}=[B\left(L\right),L]\left(t\right)$ with initial datum $L\left(0\right)=L_0$. Then, the solution to the deformed flow

$$\dot{\widehat{L}}={\displaystyle \frac{dL\left({\alpha }^{-1}t\right)}{dt}}={[B\left(L\right),L]}_{(\psi ,\varphi )}^{(i,j)}\left({\alpha }^{-1}t\right){\alpha }^{-1}$$

with the same initial datum is

$$\widehat{L}\left(t\right)=L_{\mathrm{Toda}}({\alpha }^{-1} t), \alpha =r^{ p(i+1)+q(j-1)}.$$

In particular, all classical conserved quantities (e.g., $\mathrm{tr}\left(L^k\right)$) are preserved, and the phase portrait is unchanged up to a constant time rescaling.

Proof. 

Equation (16) gives $\dot{L}=\alpha [B\left(L\right),L]\left(t\right)$. Then, we have

$${\displaystyle \frac{d\widehat{L}}{dt}}={\displaystyle \frac{dL\left({\alpha }^{-1}t\right)}{dt}}=\alpha [B\left(L\right),L]\left({\alpha }^{-1}t\right){\alpha }^{-1}=[B\left(L\right),L]\left({\alpha }^{-1}t\right),$$

with $\widehat{L}\left(0\right)=L_0$. Hence, $\widehat{L}\left(t\right)=L_{\mathrm{Toda}}\left({\alpha }^{-1}t\right)$ is the solution. □

4.2. Case II: Planar Rotations

In this case, let $\psi \left(\theta \right)=\mathcal{E}\left(R_2\left(\theta \right)\right)$; two subcases—symmetry and asymmetry—are analyzed. We claim the flow will be gauge-equivalent to classical Toda and preserve tracelessness and isospectrality in the case of symmetric choice and $(i,j)=(0,0)$.

(a)

Symmetric choice

$\varphi \left(\theta \right)=\psi \left(\theta \right)$. Then, (6) gives, for $i,j\in \mathbb{N}$,

$${[B\left(L\right),L]}_{(\psi ,\psi )}^{(i,j)}\left(\theta \right)=\psi \left(\theta \right) B\left(L\right) \psi {\left(\theta \right)}^{i+j} L \psi {\left(\theta \right)}^{-1}-\psi \left(\theta \right) L \psi {\left(\theta \right)}^{i+j} B\left(L\right) \psi {\left(\theta \right)}^{-1}.$$

(17)

In particular, when $(i,j)=(0,0)$, one can factor the outer conjugations as

$${[B\left(L\right),L]}_{(\psi ,\psi )}^{(0,0)}\left(\theta \right)=\left[\psi \left(\theta \right)B\left(L\right)\psi {\left(\theta \right)}^{-1}, \psi \left(\theta \right)L\psi {\left(\theta \right)}^{-1} \right].$$

(18)

Passing to the gauge variable $\tilde{L}=\psi {\left(\theta \right)}^{-1}L\psi \left(\theta \right)$, the flow is $\dot{\tilde{L}}=[ B\left(L\right), L ]$ (Section 5). Therefore, it is isospectral and preserves the symmetry of $\tilde{L}$.

Proposition 2

(Traces and equilibria in the $2\times 2$ block). In the $2\times 2$ block with $L=\left(\begin{array}{cc}a& c\\ c& b\end{array}\right)$, one has

 1. 

$tr\left({[B\left(L\right),L]}_{(\psi ,\psi )}^{(0,0)}\left(\theta \right)\right)=0$ for all θ;

 2. 

The right-hand side of (17) vanishes identically when

$$\theta =\left\{\begin{array}{c}{\displaystyle \frac{\pi }{2}}+k\pi , i+j \mathit{is} \mathit{odd},k\in \mathbb{Z},\hfill \\ {\displaystyle \frac{\pi }{4}}+{\displaystyle \frac{k\pi }{2}}, i+j \mathit{is} \mathit{even} \mathit{and} i+j\ne 0,k\in \mathbb{Z};\hfill \end{array}\right.$$

then, every symmetric L is an equilibrium at those parameter values.

Proof. 

1 follows from (18), and $tr[X,Y]=0$. For 2, a direct $2\times 2$ computation (Appendixes Appendix B and Appendix E) shows that both diagonal and off-diagonal entries vanish at

$$\theta =\left\{\begin{array}{c}{\displaystyle \frac{\pi }{2}}+k\pi , i+j \mathrm{is} \mathrm{odd},k\in \mathbb{Z},\hfill \\ {\displaystyle \frac{\pi }{4}}+{\displaystyle \frac{k\pi }{2}}, i+j \mathrm{is} \mathrm{even} \mathrm{and} i+j\ne 0,k\in \mathbb{Z}.\hfill \end{array}\right.$$

□

Remark 3.

Notice that when $(i,j)=(0,0)$, the flow ${[B\left(L\right),L]}_{(\psi ,\psi )}^{(0,0)}\left(\theta \right)$ is traceless (as shown in Proposition 2) and isospectral. On the other hand, when $(i,j)\ne (0,0)$, the flow ${[B\left(L\right),L]}_{(\psi ,\psi )}^{(i,j)}\left(\theta \right)$ remains traceless (see Appendix B), but it is no longer isospectral.

(b)

Asymmetric choice

$\varphi \left(\theta \right)=\mathcal{E}\left(R_2(-\theta )\right)$. In this case, the $2\times 2$ block of the right-hand side becomes the trigonometric combination displayed in (A6) (see Appendix B for the derivation); it is symmetric and traceless in the $2\times 2$ setting. We do not claim global isospectrality for generic $\theta$.

4.3. Case III: Planar Hyperbolic Rotations

In this case, let $\psi \left(\lambda \right)=\mathcal{E}\left(H_2\left(\lambda \right)\right)$. The flow is then gauge-equivalent to the classical Toda flow and preserves tracelessness and isospectrality in the symmetric choice with $(i,j)=(0,0)$. In the case of asymmetry, the isospectrality will be broken, and the genuinely new dynamics is induced, which is not present in classical Toda.

(a)

Symmetric choice

$\varphi \left(\lambda \right)=\psi \left(\lambda \right)$. As shown in (17), for $(i,j)=(0,0)$, the deformed right-hand side can be written in the form (18) with $\theta$ replaced by $\lambda$; then, it is traceless. In the gauge variable $\tilde{L}=\psi {\left(\lambda \right)}^{-1}L\psi \left(\lambda \right)$, one has $\dot{\tilde{L}}=[ B\left(L\right), L ]$, which is an isospectral Lax equation. However, when $(i,j)\ne (0,0)$, the deformed right-hand side remains traceless, but the isospectrality no longer holds.

(b)

Asymmetric choice

$\varphi \left(\lambda \right)=\mathcal{E}\left(H_2(-\lambda )\right)$. In the $2\times 2$ block with $L=\left(\begin{array}{cc}a& c\\ c& b\end{array}\right)$, the right-hand side can be written explicitly as in (A8) (see Appendix B). A key obstruction appears at the level of traces.

Proposition 3.

For the asymmetric hyperbolic choice and generic $(a,b,c,\lambda )$, one has

$$tr\left({[B\left(L\right),L]}_{(\psi ,\varphi )}^{(i,j)}\left(\lambda \right)\right)\ne 0 \mathit{whenever} sinh\left((i-j+2)\lambda \right)\ne 0 \mathit{and} c (a-b)\ne 0.$$

(19)

Then, the deformed right-hand side cannot be written as a commutator $\left[M\right(L),L]$ with any matrix $M\left(L\right)$, and the evolution is not isospectral in general.

Proof. 

The explicit $2\times 2$ formula shows that the two diagonal entries do not sum to zero unless $sinh\left((i-j+2)\lambda \right)=0$ or $c(a-b)=0$. Since $tr[X,Y]=0$ for all $X,Y$, a nonzero trace precludes a commutator representation. □

Corollary 1.

In the asymmetric hyperbolic case, the standard Toda integrals $\mathrm{tr}\left(L^k\right)$ are not preserved for generic initial data and parameters. The parameter value $\lambda =0$ is a degenerate case returning to the undeformed bracket.

Remark 4.

The symmetric choices $(\varphi =\psi )$ and the restriction $(i,j)=(0,0)$ lead, in the gauge variable $\tilde{L}={\psi }^{-1}L\psi$, to the Toda-type Lax equation $\dot{\tilde{L}}=[ B\left(L\right),L ]$, which is isospectral. Unless the conjugation by ψ preserves the upper/lower triangular splitting that defines $B(·)$, the flow in the original variable L need not be a literal conjugate of the classical Toda evolution (see Section 5). The asymmetric choices exhibit genuinely new dynamics and, in the hyperbolic case, fail to be of commutator type for generic parameters.

The following table summarizes which deformation families preserve or break isospectrality.

Deformation Families	Parameters	Isospectrality
Classical Toda	$\varphi =\psi =Id$	Preserved
Scalar Dilations	$\psi \left(r\right)=r^p{I}_n$,$\varphi \left(r\right)=r^q{I}_n$	Preserved
Symmetric Rotations	$$\begin{gathered} \varphi \left(\theta \right)=\psi \left(\theta \right)= \\ \mathcal{E}\left(R_2\left(\theta \right)\right),(i,j)=(0,0) \end{gathered}$$	Preserved
Symmetric Rotations	$$\begin{gathered} \varphi \left(\theta \right)=\psi \left(\theta \right)= \\ \mathcal{E}\left(R_2\left(\theta \right)\right),(i,j)\ne (0,0) \end{gathered}$$	Broken
Asymmetric Rotations	$\varphi \left(\theta \right)=\mathcal{E}\left(R_2(-\theta )\right)$	Broken
Symmetric Hyperbolic	$\varphi \left(\lambda \right)=\psi \left(\lambda \right)=\mathcal{E}\left(H_2\left(\lambda \right)\right)$, $(i,j)=(0,0)$	Preserved
Symmetric Hyperbolic	$\varphi \left(\lambda \right)=\psi \left(\lambda \right)=\mathcal{E}\left(H_2\left(\lambda \right)\right)$, $(i,j)\ne (0,0)$	Broken
Asymmetric Hyperbolic	$\varphi \left(\lambda \right)=\mathcal{E}\left(H_2(-\lambda )\right)$	Broken

5. Miura-Type Transformations

In the symmetric setting $\varphi \left(s\right)=\psi \left(s\right)$ and $(i,j)=(0,0)$, the bracket (6) takes the concrete form

$${[A,B]}_{(\psi ,\psi )}^{(0,0)}\left(s\right)=\psi \left(s\right) AB \psi {\left(s\right)}^{-1}-\psi \left(s\right) BA \psi {\left(s\right)}^{-1}.$$

A convenient way to analyze the flow is to use a gauge map together with a conjugated triangular projection.

Fix $s\in R_1$, set

$${\mathsf{\Phi }}_{\psi }: GL\left(V\right)\to GL\left(V\right), {\mathsf{\Phi }}_{\psi }\left(L\right):=\psi {\left(s\right)}^{-1}L \psi \left(s\right),$$

(20)

and define the conjugated upper/lower-triangular projection

$$B_{\psi }\left(X\right):=\psi \left(s\right) B\left(X\right) \psi {\left(s\right)}^{-1}, B\left(X\right)=X_{>0}-X_{<0}.$$

(21)

Theorem 2

(Conditional gauge reduction). Let $\varphi \left(s\right)=\psi \left(s\right)$ and $(i,j)=(0,0)$, and consider the deformed Lax equation

$$\dot{L}={[B\left(L\right),L]}_{(\psi ,\psi )}^{(0,0)}\left(s\right).$$

With $\tilde{L}:={\mathsf{\Phi }}_{\psi }\left(L\right)$, one has the following equivalence:

$$\dot{\tilde{L}}=[ B\left(L\right), L ]\left(s\right) ⟺ B \left({\mathsf{\Phi }}_{\psi }\left(X\right)\right)=B_{{\psi }^{-1}}\left(X\right) \mathit{for} \mathit{all} X\in GL\left(V\right).$$

In particular, the gauge variable $\tilde{L}$ satisfies a Toda-type Lax equation with the projection $B_{\psi }$(then, $B_{\psi }\left(\tilde{L}\right)=B\left(L\right)$) if and only if ψ preserves the upper/lower triangular splitting (e.g., ψ is diagonal). In the absence of this symmetry, no such global reduction holds in general.

Proof. 

We have

$$\begin{array}{cc}\hfill \dot{\tilde{L}}& ={\displaystyle \frac{d{\mathsf{\Phi }}_{\psi }\left(L\right)}{dt}}\hfill \\ \hfill & ={[B\left({\mathsf{\Phi }}_{\psi }\left(L\right)\right),{\mathsf{\Phi }}_{\psi }\left(L\right)]}_{(\psi ,\psi )}^{(0,0)}\hfill \\ \hfill & ={[B\left(\psi {\left(s\right)}^{-1}L\psi \left(s\right)\right),\psi {\left(s\right)}^{-1}L\psi \left(s\right)]}_{(\psi ,\psi )}^{(0,0)}\hfill \\ \hfill & =\psi \left(s\right)B\left(\psi {\left(s\right)}^{-1}L\psi \left(s\right)\right)\psi {\left(s\right)}^{-1}L-L\psi \left(s\right)B\left(\psi {\left(s\right)}^{-1}L\psi \left(s\right)\right)\psi {\left(s\right)}^{-1}\hfill \\ \hfill & =[\psi \left(s\right)B\left(\psi {\left(s\right)}^{-1}L\psi \left(s\right)\right)\psi {\left(s\right)}^{-1},L].\hfill \end{array}$$

Thus, $\dot{\tilde{L}}=[ B\left(L\right), L ]$ if and only if $B\left(L\right)=\psi \left(s\right)B\left(\psi {\left(s\right)}^{-1}L\psi \left(s\right)\right)\psi {\left(s\right)}^{-1}$, i.e., $B \left({\mathsf{\Phi }}_{\psi }\left(L\right)\right)=B_{{\psi }^{-1}}\left(L\right)$ for any $L\in GL\left(V\right)$. □

Remark 5.

Typical instances where the splitting is preserved include scalar dilations $\psi =r^p{I}_n$ and diagonal sign/permutation matrices. For generic dense rotations or hyperbolic rotations, the splitting is not preserved; thus a direct gauge reduction to a Toda-type commutator equation fails in general.

We next record blockwise Miura-type relations on embedded $n\times n$ blocks, which are independent of the global splitting property and will be used for explicit formulas.

For the symmetric block $L={\left(a_{ij}\right)}_{n\times n}$ and the rotation $\mathcal{E}\left(R_2\left(\theta \right)\right)=\mathrm{diag}(I_{n-2},R_2\left(\theta \right))$, we set

$${\mathcal{M}}_{\theta }\left(L\right):=\mathcal{E}\left(R_2\left(\theta \right)\right) L \mathcal{E}{\left(R_2\left(\theta \right)\right)}^{-1}.$$

Then, the entries $\{{\mathbf{a}}_{ij} | 1\le i,j\le n\}$ of ${\mathcal{M}}_{\theta }\left(L\right)$ are

$$\begin{array}{cc}\hfill {\mathbf{a}}_{ij}& =a_{ij}, 1\le i,j\le n-2,\hfill \\ \hfill {\mathbf{a}}_{in-1}& =a_{in-1}cos\left(\theta \right)-a_{in}sin\left(\theta \right), 1\le i\le n-2,\hfill \\ \hfill {\mathbf{a}}_{n-1i}& =a_{in-1}cos\left(\theta \right)-a_{in}sin\left(\theta \right), 1\le i\le n-2,\hfill \\ \hfill {\mathbf{a}}_{in}& =a_{in-1}sin\left(\theta \right)+a_{in}cos\left(\theta \right), 1\le i\le n-2,\hfill \\ \hfill {\mathbf{a}}_{ni}& =a_{in-1}sin\left(\theta \right)+a_{in}cos\left(\theta \right), 1\le i\le n-2,\hfill \\ \hfill {\mathbf{a}}_{n-1n-1}& =a_{n-1n-1}{cos}^2\theta +a_{nn}{sin}^2\theta -a_{n-1n}sin\left(2\theta \right),\hfill \\ \hfill {\mathbf{a}}_{nn}& =a_{n-1n-1}{sin}^2\theta +a_{nn}{cos}^2\theta +a_{n-1n}sin\left(2\theta \right),\hfill \\ \hfill {\mathbf{a}}_{n-1n}& =\frac{1}{2}(a_{n-1n-1}-a_{nn})sin\left(2\theta \right)+a_{n-1n}cos\left(2\theta \right),\hfill \\ \hfill {\mathbf{a}}_{nn-1}& =\frac{1}{2}(a_{n-1n-1}-a_{nn})sin\left(2\theta \right)+a_{n-1n}cos\left(2\theta \right).\hfill \end{array}$$

(22)

Similarly, for the hyperbolic rotation $\mathcal{E}\left(H_2\left(\lambda \right)\right)=\mathrm{diag}(I_{n-2},H_2\left(\lambda \right))$, we define

$${\mathcal{N}}_{\lambda }\left(L\right):=\mathcal{E}\left(H_2\left(\lambda \right)\right) L \mathcal{E}{\left(H_2\left(\lambda \right)\right)}^{-1}.$$

Its entries $\{{\widehat{a}}_{ij} | 1\le i,j\le n\}$ are

$$\begin{array}{cc}\hfill {\widehat{a}}_{ij}& =a_{ij}, 1\le i,j\le n-2,\hfill \\ \hfill {\widehat{a}}_{in-1}& =a_{in-1}cosh\lambda -a_{in}sinh\lambda , 1\le i\le n-2,\hfill \\ \hfill {\widehat{a}}_{n-1i}& =a_{in-1}cosh\lambda +a_{in}sinh\lambda , 1\le i\le n-2,\hfill \\ \hfill {\widehat{a}}_{in}& =-a_{in-1}sinh\lambda +a_{in}cosh\lambda , 1\le i\le n-2,\hfill \\ \hfill {\widehat{a}}_{ni}& =a_{in-1}sinh\lambda +a_{in}cosh\lambda , 1\le i\le n-2,\hfill \\ \hfill {\widehat{a}}_{n-1n-1}& =a_{n-1n-1}{cosh}^2\lambda -a_{nn}{sinh}^2\lambda ,\hfill \\ \hfill {\widehat{a}}_{nn}& =-a_{n-1n-1}{sinh}^2\lambda +a_{nn}{cosh}^2\lambda ,\hfill \\ \hfill {\widehat{a}}_{n-1n}& =a_{n-1n}+\frac{1}{2}(a_{nn}-a_{n-1n-1})sinh\left(2\lambda \right),\hfill \\ \hfill {\widehat{a}}_{nn-1}& =a_{n-1n}+\frac{1}{2}(a_{n-1n-1}-a_{nn})sinh\left(2\lambda \right).\hfill \end{array}$$

(23)

Proposition 4

(Blockwise Miura conjugacy in $n\times n$). Fix $\varphi =\psi$ and $(i,j)=(0,0)$. Let $\mathcal{F}$ denote the $n\times n$ classical Toda vector field induced by $\dot{L}=[B\left(L\right),L]$ on the symmetric block ${\left(a_{ij}\right)}_{n\times n}$ and ${\mathcal{F}}_{\theta }$(resp. ${\mathcal{F}}_{\lambda }$) denote the $n\times n$ deformed vector field coming from the rotational(resp. hyperbolic) case of Section 4; then,

$$\dot{\mathbf{u}}={\mathcal{F}}_{\theta }\left(u\right) ⟺ \dot{u}=\mathcal{F}\left(u\right) \mathit{with} \mathbf{u}={\mathcal{M}}_{\theta }u,$$

$$\dot{\widehat{u}}={\mathcal{F}}_{\lambda }\left(u\right) ⟺ \dot{u}=\mathcal{F}\left(u\right) \mathit{with} \widehat{u}={\mathcal{N}}_{\lambda }u,$$

where $u={\left(a_{ij}\right)}_{n\times n}$, $\mathbf{u}={\left({\mathbf{a}}_{ij}\right)}_{n\times n}$, and $\widehat{u}={\left({\widehat{a}}_{ij}\right)}_{n\times n}$. Equivalently,

$${\mathcal{M}}_{\theta }\circ \mathcal{F}={\mathcal{F}}_{\theta }, {\mathcal{N}}_{\lambda }\circ \mathcal{F}={\mathcal{F}}_{\lambda };$$

that is, the Miura transforms ${\mathcal{M}}_{\theta }$ and ${\mathcal{N}}_{\lambda }$ intertwine the undeformed and deformed vector fields on the $n\times n$ block.

Proof. 

A direct verification is provnd in Appendix D. □

Remark 6.

The classical Toda system is recovered from the deformed model in two transparent ways on the $n\times n$ block: (i) by taking the undeformed parameter ($\theta =0$ or $\lambda =0$), where ${\mathcal{M}}_{\theta }$ and ${\mathcal{N}}_{\lambda }$ reduce to the identity; (ii) in settings where the splitting symmetry is preserved so that the gauge reduction in Theorem 2 applies. In particular, spectral data are preserved in case (ii).

6. Weakly Coupled Toda Lattices with Indefinite Metrics

We study the weakly coupled (finite, nonperiodic) Toda lattice with an indefinite signature and its deformed counterpart. Our emphasis is on a clean Lax formulation, the semidirect-product structure, and a concise $2\times 2$ normal form. The sign matrix $S=\mathrm{diag}(s_1,\dots ,s_N)$ encodes an underlying metric symmetry that is preserved along the Lax flows.

Fix signs $s_k\in \{\pm 1\}$ ($k=1,\dots ,N$), let $S=\mathrm{diag}(s_1,\dots ,s_N)$. We consider real symmetric tridiagonal matrices

$$\begin{array}{cc}\hfill & L=\left(\begin{array}{ccccc}{s}_1{a}_1& s_2{b}_1& & \hfill & 0\\ s_1{b}_1& s_2{a}_2& s_3{b}_2\\ & \ddots & \ddots & \ddots \\ & & s_{N-2}{b}_{N-2}& s_{N-1}{a}_{N-1}& s_N{b}_{N-1}\\ 0& & \hfill & s_{N-1}{b}_{N-1}& s_N{a}_N\end{array}\right),\hfill \\ \hfill & \hfill \\ \hfill & \mathrm{with} \widehat{L} \mathrm{defined} \mathrm{likewise} \mathrm{with} (a_k,b_k)⇝({\widehat{a}}_k,{\widehat{b}}_k),\hfill \end{array}$$

with boundary conditions $b_0={\widehat{b}}_0=b_N={\widehat{b}}_N=0$. As before, we write $B\left(X\right):=X_{>0}-X_{<0}$, and set

$$B:=\frac{1}{4}\left(L_{>0}-L_{<0}\right), \widehat{B}:=\frac{1}{4}\left({\widehat{L}}_{>0}-{\widehat{L}}_{<0}\right).$$

(24)

Remark 7

(Normalization). The factor $\frac{1}{4}$ in (24) is a harmless coefficient normalization that simplifies the component formulas below; it amounts to a constant rescaling of time compared to the convention $B\left(X\right)=X_{>0}-X_{<0}$.

Definition 1.

The weakly coupled system is the Lax pair on $\mathfrak{T}⋉\mathfrak{T}$ (the vector space $\mathfrak{T}$ of symmetric tridiagonal matrices) given by

$$\dot{L}=[B,L], \dot{\widehat{L}}=[\widehat{B},L]+[B,\widehat{L}].$$

(25)

Remark 8.

The first component evolves autonomously by a Toda-type Lax equation, and the second component is driven linearly by L via the adjoint action together with its own projection $\widehat{B}$. This structure, referred to as weak coupling, is described by extended Toda equations where the interactions between the components are weaker and more decoupled. In contrast, in the “strong" coupling case, both components contribute to the first equation, with the transformation of variables and interactions between them becoming more entangled. This results in more complex and sensitive equations. Our formulas below clarify and correct places in the literature where these two regimes were mixed, see, e.g., [18,19].

From (25), one reads the component form (for $k=1,\dots ,N$):

$$\begin{array}{cc}\hfill {\dot{a}}_k& =\frac{1}{2}\left(s_{k+1}{b}_k^2-s_{k-1}{b}_{k-1}^2\right),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\dot{b}}_k& =\frac{1}{4} b_k\left(s_{k+1}{a}_{k+1}-s_k{a}_k\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \dot{{\widehat{a}}_k}& =s_{k+1}{b}_k{\widehat{b}}_k-s_{k-1}{b}_{k-1}{\widehat{b}}_{k-1},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \dot{{\widehat{b}}_k}& =- \frac{1}{4}\left[(s_k{a}_k-s_{k+1}{a}_{k+1}) {\widehat{b}}_k+(s_k{\widehat{a}}_k-s_{k+1}{\widehat{a}}_{k+1}) b_k\right].\hfill \end{array}$$

(29)

Remark 9.

Compared with the sign pattern in (27) and (28), the minus sign in (29) is essential (see the $2\times 2$ check below).

Proposition 5.

For the weakly coupled system (25), one has ${\displaystyle \frac{d}{dt}}tr\left(L^m\right)=0$ for all $m\ge 1$. In particular, the spectrum of L is preserved. No analogous general claim is made for mixed quantities involving $\widehat{L}$.

Proof. 

By cyclicity of the trace, we have ${\displaystyle \frac{d}{dt}}tr\left(L^m\right)=m tr\left(L^{m-1}[B,L]\right)=0$. □

6.1. Two-Dimensional Normal Form

We illustrate (25) for $N=2$ and $S=\mathrm{diag}(1,-1)$; that is,

$$L_2=\left(\begin{array}{cc}{a}_1& -b_1\\ b_1& -a_2\end{array}\right), {\widehat{L}}_2=\left(\begin{array}{cc}{\widehat{a}}_1& -{\widehat{b}}_1\\ {\widehat{b}}_1& -{\widehat{a}}_2\end{array}\right).$$

A direct computation shows that (26)–(29) become

$$\begin{array}{cc}{\dot{a}}_1\hfill & =-\frac{1}{2} b_1^2,\hspace{1em}{\dot{a}}_2=-\frac{1}{2} b_1^2,\hspace{1em}{\dot{b}}_1=-\frac{1}{4} b_1 (a_1+a_2),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\dot{{\widehat{a}}_1}\hfill & =-b_1{\widehat{b}}_1,\hspace{1em}\dot{{\widehat{a}}_2}=-b_1{\widehat{b}}_1,\hspace{1em}\dot{{\widehat{b}}_1}=-\frac{1}{4}\left[(a_1+a_2){\widehat{b}}_1+({\widehat{a}}_1+{\widehat{a}}_2)b_1\right].\hfill \end{array}$$

(31)

Here, $B_2:=\frac{1}{4}\left({\left(L_2\right)}_{>0}-{\left(L_2\right)}_{<0}\right)$, and ${\widehat{B}}_2:=\frac{1}{4}\left({\left({\widehat{L}}_2\right)}_{>0}-{\left({\widehat{L}}_2\right)}_{<0}\right)$ (consistent with (24)).

Remark 10.(i). Equations (30) for $L_2$ do not contain ${\widehat{b}}_1$ and ${\widehat{a}}_k$. (ii). Any such terms would contradict the autonomous Lax equation $\dot{L}=[B,L]$. Compare [18,19].

Spectral data and classification.

For

$$L_2=\left(\begin{array}{cc}{a}_1& -b_1\\ b_1& -a_2\end{array}\right),\hspace{1em}\mathrm{tr} L_2=a_1-a_2, det{L}_2=-a_1{a}_2+b_1^2,$$

the characteristic polynomial is ${\lambda }^2-(a_1-a_2)\lambda +(-a_1{a}_2+b_1^2)=0.$ Then,

$${\lambda }_{1,2}={\displaystyle \frac{(a_1-a_2)\pm \sqrt{{(a_1-a_2)}^2-4(-a_1{a}_2+b_1^2)}}{2}}.$$

(32)

In particular, the spectrum is real if and only if the discriminant $\mathsf{\Delta }={(a_1-a_2)}^2-4(-a_1{a}_2+b_1^2)$ is nonnegative. Any classification purely in terms of $m:=a_2-a_1$ is valid only after a normalization such as $b_1\equiv 1$ and $a_1+a_2\equiv 0$, in which case (32) reduces to ${\lambda }_{1,2}=\frac{1}{2}(-m\pm \sqrt{m^2-4}).$ We shall work with the invariants $\mathrm{tr} L_2$ and $det{L}_2$ in the general case.

For the second component, (31) is a linear nonautonomous equation on ${\widehat{L}}_2$ of the form

$$\dot{{\widehat{L}}_2}-[B_2,{\widehat{L}}_2]=[{\widehat{B}}_2,L_2].$$

It admits the variation-of-constants representation

$${\widehat{L}}_2\left(t\right)=U\left(t\right)\left({\widehat{L}}_2\left(0\right)+{\int }_0^{t}U{\left(\tau \right)}^{-1} [{\widehat{B}}_2\left(\tau \right),L_2\left(\tau \right)] U\left(\tau \right) d\tau \right)U{\left(t\right)}^{-1},\hspace{1em}\dot{U}=B_2 U, U\left(0\right)=I,$$

(33)

which clarifies the semidirect nature of the coupling.

6.2. Deformed Weakly Coupled System and Gauge Reduction

We now replace the commutator in (25) by the BiHom-type bracket introduced in (6). In the symmetric setting $\varphi \left(s\right)=\psi \left(s\right)$ and $(i,j)=(0,0)$, the pushforward mechanism of Section 5 extends verbatim to the coupled case and yields a twisted semidirect system.

Proposition 6

(Twisted weak coupling under the gauge map). Let $\varphi \left(s\right)=\psi \left(s\right)$ and $(i,j)=(0,0)$. Define

$$\tilde{L}:=\psi {\left(s\right)}^{-1}L\psi \left(s\right),\hspace{1em}\tilde{\widehat{L}}:=\psi {\left(s\right)}^{-1}\widehat{L}\psi \left(s\right).$$

Then, $(L,\widehat{L})$ solves the deformed weakly coupled system

$$\dot{L}={[B,L]}_{(\psi ,\psi )}^{(0,0)}\left(s\right),\hspace{1em}\dot{\widehat{L}}={[\widehat{B},L]}_{(\psi ,\psi )}^{(0,0)}\left(s\right)+{[B,\widehat{L}]}_{(\psi ,\psi )}^{(0,0)}\left(s\right),$$

if and only if $(\tilde{L},\tilde{\widehat{L}})$ solves the twisted weakly coupled system

$$\left\{\begin{array}{cc}\hfill \dot{\tilde{L}} & = B\left(L\right)L-LB\left(L\right),\hfill \\ \hfill \dot{\tilde{\widehat{L}}} & = B\left(\widehat{L}\right)L-LB\left(\widehat{L}\right)+ B\left(L\right)\widehat{L}-\widehat{L}B\left(L\right).\hfill \end{array}\right.$$

Proof. 

This is identical to the proof of Theorem 2, applied separately to each bracket ${[\widehat{B},L]}_{(\psi ,\psi )}^{(0,0)}\left(s\right)$ and ${[B,\widehat{L}]}_{(\psi ,\psi )}^{(0,0)}\left(s\right)$, and using $\widehat{B}=B\left(\widehat{L}\right)$, $B=B\left(L\right)$. □

6.3. Two-Dimensional Miura-Type Formulas

In the embedded $2\times 2$ case, the gauge map of Section 5 reduces to the explicit formulas of Section 5: for a rotation $R_2\left(\theta \right)$, define ${\mathcal{M}}_{\theta }\left(L\right):=R_2\left(\theta \right) L R_2{\left(\theta \right)}^{-1}$ and, for a hyperbolic rotation $H_2\left(\lambda \right)$, set ${\mathcal{N}}_{\lambda }\left(L\right):=H_2\left(\lambda \right) L H_2{\left(\lambda \right)}^{-1}$. Then,

$$\left(\begin{array}{cc}\mathbf{a}& c\\ \mathbf{c}& \mathbf{b}\end{array}\right)={\mathcal{M}}_{\theta }\left(\begin{array}{cc}a& c\\ c& b\end{array}\right), \left(\begin{array}{cc}\widehat{a}& \widehat{c_1}\\ {\widehat{c}}_2& \widehat{b}\end{array}\right)={\mathcal{N}}_{\lambda }\left(\begin{array}{cc}a& c\\ c& b\end{array}\right),$$

with the components given by (22), and analogously for ${\mathcal{N}}_{\lambda }$ with (23). This provides a transparent blockwise Miura-type relation between solutions of the twisted weakly coupled system and those of the deformed one in the symmetric setting.

Remark 11.

If $\varphi \ne \psi$, the trace obstructions discussed in Section 4.3 reappear already at the level of the first component, and the commutator-type structure is generally lost.

7. Conclusions

In this paper, we introduced a BiHom-type deformation of the finite nonperiodic Toda lattice by constructing a skew-symmetric bracket on $GL\left(V\right)$ from two commuting inner automorphisms $\alpha ={\mathrm{Ad}}_{\psi }$ and $\beta ={\mathrm{Ad}}_{\varphi }$, indexed by integers i and j. The resulting quadruple $(GL\left(V\right),{[·,·]}_{(\psi ,\varphi )}^{(i,j)},\alpha ,\beta )$ satisfies the BiHom–Lie axioms and, when inserted into the Toda Lax equation, produces a family of deformed lattice flows that contains both the classical Toda and the known Hom–Toda models as special cases. Working out three representative one-parameter deformations (scalar dilations, planar rotations, and hyperbolic rotations), we identified the regimes in which the flow is gauge-equivalent to a Toda-type Lax equation and hence isospectral and exhibited, in particular for asymmetric hyperbolic rotations, a trace obstruction, showing that generically the right-hand side is not a commutator, and the isospectral property is broken. On embedded $2\times 2$ blocks, we derived explicit trigonometric and hyperbolic formulae and Miura-type transformations which make symmetry constraints (such as tracelessness and conjugation invariance) completely transparent.

We further formulated a weakly coupled Toda lattice with an indefinite metric and its BiHom-type deformation, expressed the dynamics as a semidirect Lax system, and derived a $2\times 2$ normal form together with explicit inverse-scattering-type solutions, thereby clarifying and correcting certain evolution formulas that appear in the literature. The present construction is not exhaustive: the uniqueness of our bracket is proved only within a specific bilinear ansatz, and more general BiHom–Lie deformations of Toda-type systems remain to be investigated. Likewise, a complete spectral and asymptotic theory for the asymmetric (non-isospectral) cases is still open, despite supporting evidence from explicit low-dimensional examples and symbolic computations.