Integrability of a Family of Four-Dimensional Quadratic Lotka–Volterra Complex Systems

Abstract

This work addresses the integrability of four-dimensional quadratic Lotka–Volterra systems exhibiting (1:−1:i:−i) resonance by identifying the parameter constraints required for such properties to exist. Our results show that, within this family, integrability is connected to linearizability and time-reversibility. To establish these results, we combine theoretical and computational methods. On the theoretical side, we use Poincaré–Dulac normal form theory and methods based on Gröbner basis theory. On the computational side, we apply the algorithms described in the paper to compute normal forms and minimal associated prime ideals required for the analysis of the integrability conditions.

1. Introduction

Characterizing the integrability properties of a system is of high importance in qualitative differential equation theory. This area of study goes back to the classic research of Poincaré and Darboux in the late nineteenth century, where geometric and algebraic methods were first applied to analyze polynomial vector fields [1,2,3,4]. Since that time, various methods have been developed, such as the Darboux theory of integrability and the Poincaré–Dulac normal form theory [1,5,6]. A closely related problem involves determining the linearizability of singular points in polynomial systems (see, for example, [7,8,9] and the references therein). These subjects continue to be actively studied today because of their importance in several applications.

One standard way to analyze if a nonlinear system is locally integrable near an equilibrium is to use normal form theory to simplify the equations. In this work, we study both formal and analytic first integrals using the Poincaré–Dulac normal form framework (see, e.g., [2,7,10,11,12]). Standard results in normal form theory ensure that an invertible formal change of coordinates can reduce any formal vector field to its Poincaré–Dulac normal form. Notably, such a transformation preserves the formal first integrals of the system.

Much of the existing literature on the integrability of polynomial systems focuses on the planar case. Dulac in [13] classified integrable systems with (1:−1) resonance having quadratic nonlinearities. Later, the authors in [14] studied the integrability of quadratic singularities with (1:−2) resonance. Many other researchers have looked at (p:−q) resonances for critical points (see, e.g., [8,15,16,17]). More general results for planar vector field integrability can be found in works like [18,19,20,21].

Integrability has also been a major topic for three-dimensional differential systems. As the dimension and the degree of the nonlinear terms increase, the problem becomes much harder, leading many researchers to focus on specific classes of systems. A well-known family is the Lotka–Volterra systems (see, e.g., [22,23,24]). In [25], the authors found a full set of conditions for integrability for quadratic Lotka–Volterra systems with (1:−1:1), (2:−1:1) and (1:−2:1) resonances at the origin. The (1:−1:−1) resonance case for 3D quadratic systems was studied in [26,27]. Other quadratic 3D classes have also been explored, for instance, in [28,29].

There has been much less research into the local integrability of four-dimensional systems. Specifically, we mention [30,31], which investigated quadratic systems with (1:−1:1:1) and (1:−1:1:−1) resonances.

We focus here on a specific family of four-dimensional, degree-two polynomial systems characterized by (1:−1:i:−i) resonance, described by the equations:

$$\begin{array}{cc}\hfill \dot{x}=& x(1+a_{1000}x+a_{0100}y+a_{0010}z),\hfill \\ \hfill \dot{y}=& y(-1+b_{0100}y+b_{1000}x+b_{0001}w),\hfill \\ \hfill \dot{z}=& z(i+c_{0100}y+c_{0010}z+c_{0001}w),\hfill \\ \hfill \dot{w}=& w(-i+d_{1000}x+d_{0010}z+d_{0001}w),\hfill \end{array}$$

(1)

where $x,y,z,w$ are complex unknown functions, the parameters $a_k,b_k,c_k,d_k$ are complex numbers, and $i=\sqrt{-1}$. We want to find the conditions on these parameters such that (1) has two independent analytic first integrals in a neighborhood of the origin.

Moser [32] studied Hamiltonian systems near an equilibrium which are of the form:

$$\begin{array}{cc}\hfill {\dot{x}}_{\nu }& =H_{y_{\nu }}(x,y),\hfill \\ \hfill {\dot{y}}_{\nu }& =-H_{x_{\nu }}(x,y),\hfill \end{array}$$

where $\nu \in \{1,2,\dots ,n\}$; for general eigenvalues $(\pm {\alpha }_1,\dots ,\pm {\alpha }_n)$ where the $2n$ complex numbers are distinct, ${\alpha }_1$ and ${\alpha }_2$ are independent over the reals; and ${\alpha }_j\ne m_1{\alpha }_1+m_2{\alpha }_2$ for all integers $m_1,m_2$ and $j\ge 3$. He showed that although these systems often have ‘small divisor’ divergence issues, they still have a four-parameter family of convergent solutions. The resonance in our system (1) is a specific instance of the case Moser considered. In Section 3 of [32], it is shown that these Hamiltonian systems satisfy $a_{\kappa }+b_{\kappa }=0$, which is equivalent to having two independent analytic first integrals near the origin.

Recall that a dynamical system $\dot{\mathbf{x}}=F\left(\mathbf{x}\right)$ is called time-reversible if there is an involution $\phi$ satisfying:

$$D_{\phi }^{-1}·F\circ \phi =-F.$$

(2)

Time-reversal symmetry is a fundamental property found in many natural systems. For example, it appears in many models for classical and quantum mechanics. Because it has such a strong effect on the dynamics, it has been extensively researched within the framework of differential equations (e.g., [33,34,35,36]). Also, the relationship between local integrability and time-reversal symmetry has been examined in [10,37,38].

The structure of this paper is as follows: Section 2 provides the fundamental definitions and preliminary results. In Section 3, we describe the theoretical tools used to prove the integrability of the system. Section 4 discusses time-reversible systems and their connection to integrability. In Section 5, we find the parameter values for which subfamilies of (1) have two independent analytic first integrals or are analytically linearizable.

Although we show that first integrals exist, we do not provide them in closed form. Based on the recurrence relations for the coefficients and the structure of the normal form, it is likely that these first integrals involve special functions, such as those of hypergeometric type. This behavior was also noted in [14], where first integrals were represented by hypergeometric functions. This suggests a link between integrability, resonance, and special functions that we hope to investigate in the future.

2. Preliminaries

This section provides a review of the fundamental definitions and methodologies regarding the integrability and linearizability problem.

Consider the class of analytic systems represented by:

$$\dot{\mathbf{x}}=F\left(\mathbf{x}\right)=A\mathbf{x}+\tilde{F}\left(\mathbf{x}\right),$$

(3)

where $A=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$ is a diagonal matrix and $\mathbf{x}=(x_1,\dots ,x_n)\in {\mathbb{C}}^n$. The nonlinear part is given by $\tilde{F}\left(\mathbf{x}\right)={({\tilde{F}}_1\left(\mathbf{x}\right),\dots ,{\tilde{F}}_n\left(\mathbf{x}\right))}^{\top }$. These nonlinearities $F_i$ are power series vanishing at the origin along with their first derivatives. Let $\mathcal{X}$ be the vector field induced by (3). A non-constant, differentiable function (which may be a formal power series) $\mathsf{\Psi }$ is called a first integral of the system if it satisfies:

$$\mathcal{X}\mathsf{\Psi }=\sum _{i=1}^n{\displaystyle \frac{\partial \mathsf{\Psi }}{\partial x_i}}{\dot{x}}_i=0.$$

(4)

It is a standard result (see [10]) that a formal, invertible coordinate transformation:

$$\mathbf{x}=\mathbf{y}+\sum _{j=2}^{\infty }{h}_j\left(\mathbf{y}\right),$$

(5)

where $\mathbf{y}=(y_1,\dots ,y_n)$ and $h_j\left(\mathbf{y}\right)$ are homogeneous degree-j polynomials, (3) is converted into a system of the same general form:

$$\dot{\mathbf{y}}=A\mathbf{y}+G\left(\mathbf{y}\right),G\left(\mathbf{y}\right)=\sum _{j=2}^{\infty }{G}_j\left(\mathbf{y}\right),$$

(6)

where $G_j\left(\mathbf{y}\right)$ denotes a vector of degree-j polynomials.

Let $e_k$ represent the k-th unit basis vector in ${\mathbb{C}}^n$, $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}_0$ and $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$ the vector of eigenvalues of A in (3). We call a monomial term ${\mathbf{x}}^{\alpha }{e}_k=x_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}{e}_k$ resonant if:

$$\sum _{i=1}^n{\alpha }_i{\lambda }_i-{\lambda }_k=0$$

(7)

under the condition that ${\sum }_{i=1}^n{\alpha }_i\ge 2$.

System (6) is in Poincaré–Dulac normal form if $G\left(\mathbf{y}\right)$ contains exclusively resonant terms. The existence of a transformation (5) that transforms system (3) into this form is guaranteed by the Poincaré–Dulac theorem.

If, through a change of variables (5), we can rewrite (3) as follows:

$$\dot{\mathbf{y}}=(1+g\left(\mathbf{y}\right))A\mathbf{y}$$

(8)

for some scalar function g, then a simple time reparametrization $d\tau =(1+g(\mathbf{y}\left)\right)dt$ yields the linear system $d\mathbf{y}/d\tau =A\mathbf{y}$. Because both the linear system and (8) possess an identical number of independent polynomial first integrals, we adopt the following definition.

Definition 1.

System (3) is (completely) integrable at the origin if there is an analytic transformation (5) that transforms it to the form (8). If this can be achieved with $g\left(\mathbf{y}\right)\equiv 0$, the system is said to be linearizable.

Remark 1.

When a system’s normal form appears as (8), it is described as satisfying Bruno’s simplified condition A.

Let I be an ideal in the polynomial ring $\mathbb{K}[x_1,\dots ,x_n]$. The variety $\mathbf{V}\left(I\right)$ of the ideal I is the set of all common zeros of the polynomials in I. It can be represented as an irreducible union component:

$$\mathbf{V}={\mathbf{V}}_1\cup {\mathbf{V}}_2\cup \cdots \cup {\mathbf{V}}_m,$$

(9)

where each subset ${\mathbf{V}}_j$ is an irreducible variety. This radical $\sqrt{I}$ of I can be expressed as an intersection of prime ideals:

$$\sqrt{I}=\bigcap _{j=1}^m{Q}_j,$$

where each component ${\mathbf{V}}_j$ in the decomposition (9) coincides with the variety corresponding to one of the ideals $Q_j$. The ideals $Q_j$ are referred to as the minimal associated primes of I. The variety $\mathbf{V}\left(I\right)$ of the ideal I is then decomposed as follows:

$$\mathbf{V}\left(I\right)=\mathbf{V}\left(\sqrt{I}\right)=\bigcup _{j=1}^m\mathbf{V}\left(Q_j\right).$$

Several algorithmic approaches for determining the minimal associated primes can be found in [39,40]. They typically rely on Gröbner basis theory and are therefore extremely time and memory-consuming. In this work, we employ the routine minAssGTZ [41] implemented in the computer algebra system Singular [42], which is based on the algorithm of [39].

To perform the elimination of variables, we use a result from computational commutative algebra called the elimination theorem, which is described in what follows.

Let I be an ideal in $\mathbb{K}[x_1,\dots ,x_n]$ and m a fixed number from the set $\{0,1,\dots ,n-1\}$. The m-th elimination ideal of I is the ideal $I^{\left(m\right)}=I\cap \mathbb{K}[x_{m+1},\dots ,x_n]$. According to the elimination theorem (see, e.g., [19,43]), to compute (for any $0\le m\le n-1$) the m-th elimination ideal $I^{\left(m\right)}$ of an ideal I of $\mathbb{K}[x_1,\dots ,x_n]$, one can choose the lexicographic term order on the ring $\mathbb{K}[x_1,\dots ,x_n]$ with $x_1>x_2>\cdots >x_n$ and compute the Gröbner basis G for the ideal I with respect to that order. Then, by the elimination theorem, the set $G_m=G\cap \mathbb{K}[x_{m+1},\dots ,x_n]$ is a Gröbner basis of the m-th elimination ideal $I^{\left(m\right)}$.

3. Methodology

The objective of this section is to describe our approach for identifying the subfamilies of (1) that are integrable. Based on Poincaré–Dulac theory, system (1) can be expressed in the formal normal form:

$$\begin{array}{cc}\hfill & {\dot{y}}_1=y_1+y_1{g}_1(y_1{y}_2,y_3{y}_4),\hfill \\ \hfill & {\dot{y}}_2=-y_2+y_2{g}_2(y_1{y}_2,y_3{y}_4),\hfill \\ \hfill & {\dot{y}}_3=i{y}_3+y_3{g}_3(y_1{y}_2,y_3{y}_4),\hfill \\ \hfill & {\dot{y}}_4=-i{y}_4+y_4{g}_4(y_1{y}_2,y_3{y}_4),\hfill \end{array}$$

(10)

where $g_i$ are formal power series. The next proposition arises from the theory in [7], with an alternative derivation available in [44]. This result provides the necessary conditions for the system’s integrability.

Proposition 1.

Suppose that in system (10):

$$g_1+g_2=0,g_3+g_4=0.$$

(11)

Under these conditions, system (10) possesses two independent analytic first integrals.

In [26], the author established lemmas to address the linearizability of three-dimensional systems with integer resonances. Here, we extend this result to the case of complex resonances.

Lemma 1.

Consider the following system:

$${\dot{X}}_1=X_1,{\dot{X}}_2=i{X}_2,{\dot{x}}_3=x_3(-1+B(X_1,X_2)+c{x}_3),$$

where B is a function dependent on $X_1$ and $X_2$, and c is an arbitrary parameter. Then, the third equation can be transformed into a linear equation if we can find functions f and g such that:

$$\begin{array}{cc}\hfill & \dot{f}=B(X_1,X_2)f,\hfill \\ \hfill & \dot{g}-g=cf.\hfill \end{array}$$

(12)

Proof. 

Set $X_3=x_3{(f+g{x}_3)}^{-1}$. The third equation will be linearized if f and g are such that:

$$\mathcal{X}(f+g{x}_3)=(B(X_1,X_2)+c{x}_3)(f+g{x}_3)$$

(13)

Indeed, we have:

$$\begin{array}{cc}\hfill {\dot{X}}_3& ={\dot{x}}_3{(f+g{x}_3)}^{-1}-x_3{(f+g{x}_3)}^{-2}\mathcal{X}(f+g{x}_3)\hfill \\ \hfill & =x_3(-1+B(X_1,X_2)+c{x}_3){(f+g{x}_3)}^{-1}-x_3{(f+g{x}_3)}^{-2}(B(X_1,X_2)+c{x}_3)(f+g{x}_3)\hfill \\ \hfill & =x_3{(f+g{x}_3)}^{-1}(-1+B(X_1,X_2)+c{x}_3)-x_3{(f+g{x}_3)}^{-1}(B(X_1,X_2)+c{x}_3)\hfill \\ \hfill & =-x_3{(f+g{x}_3)}^{-1}=-X_3.\hfill \end{array}$$

For f and g to be such that (13) holds, we have the following:

The left-hand side of (13) is:

$$\begin{array}{cc}\hfill \mathcal{X}(f+g{x}_3)& =\dot{f}+\dot{g}{x}_3+g{\dot{x}}_3=\dot{f}+\dot{g}{x}_3+g{x}_3(-1+B(X_1,X_2)+c{x}_3)\hfill \\ \hfill & =\dot{f}+\dot{g}{x}_3-g{x}_3+B(X_1,X_2)g{x}_3+cg{x}_3^2.\hfill \end{array}$$

The right-hand side of (13) is:

$$\begin{array}{c}\hfill (B(X_1,X_2)+c{x}_3)(f+g{x}_3)=B(X_1,X_2)f+B(X_1,X_2)g{x}_3+cf{x}_3+cg{x}_3^2.\end{array}$$

These two expressions will be equal if f and g are such that $\dot{f}=B(X_1,X_2)f$ and $\dot{g}-g=cf$, which is exactly what wanted to show.    □

Lemma 2.

Consider the following system:

$${\dot{X}}_1=X_1,{\dot{X}}_2=i{X}_2,{\dot{x}}_3=x_3(-1+B(X_1,X_2)),$$

where B is a function dependent on $X_1$ and $X_2$. Then, the third equation is linearizable if there is some function f for which $\dot{f}=-B(X_1,X_2)$.

Proof. 

Set $X_3=x_3{e}^f$ such that $\dot{f}=-B(X_1,X_2)$. Then:

$$\begin{array}{cc}\hfill {\dot{X}}_3& ={\dot{x}}_3{e}^f+x_3{e}^f\dot{f}=x_3(-1+B(X_1,X_2))e^f+x_3{e}^f\dot{f}\hfill \\ \hfill & =x_3{e}^f(-1+B(X_1,X_2)+\dot{f})=-x_3{e}^f=-X_3.\hfill \end{array}$$

   □

Remark 2.

Lemmas 1 and 2 hold also if we change the resonances to $(1$:−i:−$1)$, (−1:i:$1)$, (−1:−i:$1)$, $(1$:i:−i), (−1:i:−i), $(1$:−i:i), (−1:i:−i). The proof in these cases goes similarly.

Remark 3.

Lemmas 1 and 2 hold also when there are only two equations with resonances $(1$:$i)$, $(1$:−$i)$, $(-$1:$i)$, $(-$1:−$i)$.

A theorem by Pliss [45] addresses scenarios where, despite the eigenvalues set being resonant, the corresponding normal form remains linear.

Theorem 1.

Consider the system (3) and assume that:

(a) 

The values of ${\sum }_{i=1}^n{\alpha }_i{\lambda }_i-{\lambda }_j$ that are not zero satisfy the condition:

$$\left|\sum _{i=1}^n{\alpha }_i{\lambda }_i-{\lambda }_j\right|>C{({\alpha }_1+\cdots +{\alpha }_n)}^{-\sigma },\forall j\in \{1,\dots ,n\},$$

(b) 

For some $C,\sigma >0$, and there exists a normal form of (3) that is linear.

Then, there exists a convergent transformation that transforms the system to its linear part.

As a particular case of Lemma 2.1 of [44], the following lemma holds.

Lemma 3.

For system (1), the non-zero elements among ${\sum }_{i=1}^4{\alpha }_i{\lambda }_i-{\lambda }_j$ satisfy:

$$\left|\sum _{i=1}^4{\alpha }_i{\lambda }_i-{\lambda }_j\right|\ge 1.$$

(14)

From this lemma, we see that $|\alpha ·\lambda -{\lambda }_j|\ne 0$. This implies that the condition (a) of Theorem 1 is satisfied for $C=\sigma =1$, because:

$$|\alpha ·\lambda -{\lambda }_j|\ge 1>{({\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4)}^{-1}=C{({\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4)}^{-\sigma }.$$

By Theorem 1, this means that if (1) is linearizable by a formal substitution, it is also analytically linearizable.

The linear system $\dot{X}=X,\dot{Y}=-Y,\dot{Z}=iZ,\dot{W}=-iW$ possesses the two first integrals $XY$ and $ZW$. Consequently, by linearizing system (1) and applying the inverse coordinate transformation, we arrive at two analytic first integrals for the original system. This shows that the linearizability of (1) is a sufficient condition for its analytic integrability.

To determine the conditions on parameters which are necessary for integrability, we compute subsequently the coefficients of the normal form of (1) using the algorithm from [12] and proceed as follows. The coefficients $g_1,g_2,g_3,$ and $g_4$ in (10) are series in variables $y_1,\dots ,y_4$ whose coefficients are polynomial functions of the system parameters. Let $u=y_1{y}_2$, $v=y_3{y}_4$. We can write:

$$g_1+g_2=\sum _{i+j=1}^{\infty }{p}_{ij}{u}^i{v}^j,g_3+g_4=\sum _{i+j=1}^{\infty }{q}_{ij}{u}^i{v}^j,$$

where $p_{ij}$, $q_{ij}$ are polynomials in parameters of (1). First, we computed the normal form of (1) through order three and calculated $g_1+g_2$ and $g_3+g_4$ up to this order and obtained polynomials $p_{10},p_{01},q_{10},q_{01}$, which are as follows:

$$\begin{array}{cc}\hfill p_{10}=& b_{1000}{b}_{0100}-a_{1000}{a}_{0100};\hfill \\ \hfill p_{01}=& i{a}_{0010}{c}_{0001}-i{b}_{0001}{d}_{0010};\hfill \\ \hfill q_{10}=& b_{1000}{c}_{0100}-a_{0100}{d}_{1000};\hfill \\ \hfill q_{01}=& i{c}_{0001}{c}_{0010}-i{d}_{0001}{d}_{0010}.\hfill \end{array}$$

They generated the ideal which we denote by $I_1$, $I_1=⟨p_{ij},q_{ij}$∣$i+j\le 1⟩$. Then, we calculated the normal form up to order four, the expressions of $g_1+g_2$ and $g_3+g_4$ up to this order, and generated $I_2=⟨p_{ij},q_{ij}$∣$i+j\le 2⟩$. Using the radical membership test (see e.g., [19,43]), we checked if $\sqrt{I_1}=\sqrt{I_2}$ and obtained $\sqrt{I_1}\ne \sqrt{I_2}$. Successive computations for higher orders $n=3,4,\dots$ yield a sequence of ideals $I_n=⟨p_{ij},q_{ij}$∣$i+j\le n⟩$. On each step, we can similarly check if $\sqrt{I_{n-1}}=\sqrt{I_n}$. Thus, we have an increasing chain of ideals $\sqrt{I_1}\subseteq \sqrt{I_2}\subseteq \sqrt{I_3}\subseteq \cdots .$ We perform computations until for some $n_0$ we have $\sqrt{I_{n_0-1}}=\sqrt{I_{n_0}}$. When we find such $n_0$, we assume that the chain reaches stability at $n_0$. To prove this hypothesis, we compute the irreducible decomposition of $\mathbf{V}\left(I_{n_0}\right)$ and then prove that systems corresponding to each component admit two independent analytic first integrals near the origin. If we are able to prove this, that yields that $I_{n_0}=I_{n_0+i}$ for all $i\in \mathbb{N}$, that is, the variety of $I_{n_0}$ defines the set of all completely integrable systems in family (1).

In the case of system (1) we found that $\sqrt{I_8}=\sqrt{I_9}$. Applying the minAssGTZ routine to $I_8$ allows us to determine the minimal associated primes of I. In our case, there are 11 prime ideals $Q_j$ such that:

$$\sqrt{I_8}=\sqrt{I_9}=\bigcap _{i=1}^{11}{Q}_j.$$

(15)

The ideals $Q_j$ are written in Section 5. The varieties of these ideals give the conditions on the parameters which are necessary for integrability, and in Section 5 we will prove that they are also the sufficient conditions.

4. Time-Reversibility

This section is devoted to presenting several results on the time-reversal symmetry in systems of the form (3) and explaining how these results are used to identify time-reversible systems within family (1). We also focus on examining the connection between time-reversibility and integrability for four-dimensional complex polynomial Lotka–Volterra systems and propose an algorithm to identify all systems within a specific polynomial family that exhibit time-reversibility.

For ${\ell }_1,{\ell }_2\in \mathbb{N}$, define the finite sets:

$$S_1=\{p_j={(p_1^{\left(j\right)},p_2^{\left(j\right)},p_3^{\left(j\right)},p_4^{\left(j\right)})}^{\top }\in {\mathbb{N}}_{-1}\times {\left({\mathbb{N}}_0\right)}^3|j=1,2,\dots ,{\ell }_1,p_1^{\left(j\right)}+p_2^{\left(j\right)}+p_3^{\left(j\right)}+p_4^{\left(j\right)}\ge 1\}$$

and

$$S_2=\{q_k={(q_1^{\left(k\right)},q_2^{\left(k\right)},q_3^{\left(k\right)},q_4^{\left(k\right)})}^{\top }\in {\left({\mathbb{N}}_0\right)}^2\times {\mathbb{N}}_{-1}\times {\mathbb{N}}_0|k=1,2,\dots ,{\ell }_2,q_1^{\left(k\right)}+q_2^{\left(k\right)}+q_3^{\left(k\right)}+q_4^{\left(k\right)}\ge 1\}.$$

Let $x=(x_1,x_2,x_3,x_4)$. Next, let $s=(s_1,s_2,s_3,s_4)$ and define $x^s=x_1^{s_1}{x}_2^{s_2}{x}_3^{s_3}{x}_4^{s_4}$. We study systems of the form:

$$\begin{array}{cc}\hfill {\dot{x}}_1& =\lambda x_1+x_1\sum _{p_j\in S_1}{a}_{p_j}{x}^{p_j},\hfill \\ \hfill {\dot{x}}_2& =-\lambda x_2+x_2\sum _{p_j\in S_1}{b}_{{\widehat{p}}_j}{x}^{{\widehat{p}}_j},\hfill \\ \hfill {\dot{x}}_3& =\mu x_3+x_3\sum _{q_k\in S_2}{c}_{q_k}{x}^{q_k},\hfill \\ \hfill {\dot{x}}_4& =-\mu x_4+x_4\sum _{q_k\in S_2}{d}_{{\widehat{q}}_k}{x}^{{\widehat{q}}_k},\hfill \end{array}$$

(16)

where ${\widehat{p}}_j$ and ${\widehat{q}}_k$ stand for $T{p}_j$ and $T{q}_k$, with T being the permutation matrix:

$$T=\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right].$$

(17)

For $p\in S_1$, we call the vector $\widehat{p}$ its conjugate vector and the parameters $a_p$ and $b_{\widehat{p}}$ dual parameters. Also, for $q\in S_2$, $\widehat{q}$ is its conjugate vector, so $c_q$ and $d_{\widehat{q}}$ are said to be dual parameters. For example, if we define sets:

$$S_1=\{(1,0,0,0),(0,1,0,0),(0,0,1,0)\}\mathrm{and}{S}_2=\{(0,1,0,0),(0,0,1,0),(0,0,0,1)\}$$

with $\lambda =1$ and $\mu =i$, we obtain family (1).

Let:

$$B=\left[\begin{array}{cccc}\alpha & 0& 0& 0\\ 0& \beta & 0& 0\\ 0& 0& \gamma & 0\\ 0& 0& 0& \delta \end{array}\right]$$

(18)

be a $4\times 4$ diagonal non-singular matrix.

Lemma 4.

The transformation:

$$\mathbf{x}=B\mathbf{y},$$

(19)

with B defined in (18), brings system (16) into a time-reversible one with respect to T if and only if:

$$CF\left(\mathbf{x}\right)=-F\left(C\mathbf{x}\right),$$

(20)

where:

$$C=\left[\begin{array}{cccc}0& {\displaystyle \frac{\beta }{\alpha }}& 0& 0\\ {\displaystyle \frac{\alpha }{\beta }}& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{\delta }{\gamma }}\\ 0& 0& {\displaystyle \frac{\gamma }{\delta }}& 0\end{array}\right].$$

(21)

Proof. 

Notice that substitution (19) transforms the system (3) into:

$$\dot{\mathbf{y}}=B^{-1}F\left(B\mathbf{y}\right)=\tilde{F}\left(\mathbf{y}\right).$$

(22)

We recall that by the definition system $\dot{\mathbf{x}}=F\left(\mathbf{x}\right)$ is time-reversible if there is an involution $\phi \left(x\right)$ such that (2) holds. Note that for the matrix T defined by (17), the map $\mathbf{x}=T\mathbf{y}$ is an involution. Therefore, system (22) possesses time-reversal symmetry under the involution $\phi =T\mathbf{y}$ if and only if:

$$T^{-1}\tilde{F}\left(T\mathbf{y}\right)=-\tilde{F}\left(\mathbf{y}\right).$$

(23)

For matrix C defined in (21), direct calculations show that $B^{-1}TB=C$, which implies that (23) is equivalent to $CF\left(\mathbf{x}\right)=-F\left(C\mathbf{x}\right)$. □

According to Lemma 4, system (16) exhibits time-reversibility provided that:

$$P·F\left(\mathbf{x}\right)=-F(P·\mathbf{x}),$$

where:

$$P=\left[\begin{array}{cccc}0& {\sigma }_1& 0& 0\\ {\sigma }_2& 0& 0& 0\\ 0& 0& 0& {\sigma }_3\\ 0& 0& {\sigma }_4& 0\end{array}\right]$$

(24)

such that ${\sigma }_1{\sigma }_2={\sigma }_3{\sigma }_4=1$.

Proposition 2.

Let $\sigma =({\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4)$ such that ${\sigma }_1{\sigma }_2={\sigma }_3{\sigma }_4=1$. System (16) is time-reversible if and only if:

$$a_{p_j}=-{\sigma }^{p_j}{b}_{{\widehat{p}}_j},c_{q_k}=-{\sigma }^{q_k}{d}_{{\widehat{q}}_k}\mathrm{for}j=1,\dots ,{\ell }_1,k=1,\dots ,{\ell }_2.$$

Proof. 

Consider system (16) and matrix P from (24). If we apply equation $P·F\left(\mathbf{x}\right)=-F(P·\mathbf{x})$ to system (16), we obtain four equations. From the first two equations, one obtains conditions $a_{p_j}{\sigma }^{{\widehat{p}}_j}=-b_{{\widehat{p}}_j}$ and $a_{p_j}=-{\sigma }^{p_j}{b}_{{\widehat{p}}_j}$ for any $j=1,\dots ,{\ell }_1$, which are, because of ${\widehat{p}}_j=T{p}_j$ and ${\sigma }_1{\sigma }_2=1$, equivalent. From the last two equations, one obtains conditions $c_{q_k}{\sigma }^{{\widehat{q}}_k}=-d_{{\widehat{q}}_k}$ and $c_{q_k}=-{\sigma }^{q_k}{d}_{{\widehat{q}}_k}$ for any $k=1,\dots ,{\ell }_2$, which are also equivalent. This concludes the proof. □

Let $\mathcal{R}\subset {\mathbb{C}}^m$ denote the set representing all systems within the family (16) that possess time-reversal symmetry, where $m=2{\ell }_1+2{\ell }_2$. Let:

$$J=⟨a_{p_j}+{\sigma }^{p_j}{b}_{{\widehat{p}}_j},c_{q_k}+{\sigma }^{q_k}{d}_{{\widehat{q}}_k},{\sigma }_1{\sigma }_2-1,{\sigma }_3{\sigma }_4-1|j=1,\dots ,{\ell }_1,k=1,\dots ,{\ell }_2⟩.$$

(25)

According to Proposition 2, to determine the collection of all time-reversible systems within the parameter space of (16), we must identify the variety associated with the elimination ideal:

$$\mathcal{I}=J\cap \mathbb{C}[a,b,c,d].$$

(26)

Now, in (16), we set $\lambda =1,\mu =-i$. The following theorem follows from Proposition 11 of [38].

Theorem 2.

If system (16) with $\lambda =1,\mu =i$ is time-reversible, then for this system, there exist two independent analytic first integrals in a neighborhood of the origin.

Theorem 3.

Assume that in system (16) $\lambda =1,\mu =i$ and let $\mathcal{I}$ be the ideal defined by (26). If the parameters of system (16) belong to the variety $\mathbf{V}\left(\mathcal{I}\right)$, then the system is integrable.

Proof. 

The variety of the ideal $\mathcal{I}$ is the Zariski closure of the set of all time-reversible systems of the form (1). By Theorem 1 of [46], the set in the space of parameters of system (16), which corresponds to systems having two analytic first integrals, is an affine variety. Therefore, taking into account Theorem 2, we see that all systems of $\mathbf{V}\left(\mathcal{I}\right)$ have two independent analytic first integrals, that is, they are integrable. □

5. The Integrability Conditions

As it is described in Section 3 the varieties of ideals $Q_i$ appearing in decomposition (15) give necessary conditions for integrability of system (1). The ideals are as follows:

$$\begin{array}{cc}\hfill Q_1& =⟨d_{0010},c_{0001},c_{0100},b_{1000},a_{0010},a_{0100}⟩;\hfill \\ \hfill Q_2& =⟨d_{0010},c_{0010},b_{1000},a_{0010},a_{0100}⟩;\hfill \\ \hfill Q_3& =⟨d_{0010},d_{1000},c_{0001},b_{1000},a_{0010},a_{0100}⟩;\hfill \\ \hfill Q_4& =⟨d_{0010},c_{0001},c_{0100},b_{0100},a_{0100}⟩;\hfill \\ \hfill Q_5& =⟨d_{0010},d_{1000},c_{0001},b_{0001},b_{1000},a_{0100}⟩;\hfill \\ \hfill Q_6& =⟨d_{0001},c_{0001},b_{0001},b_{1000},a_{0100}⟩;\hfill \\ \hfill Q_7& =⟨d_{0010},c_{0001},c_{0100},b_{0001},b_{1000},a_{0100}⟩;\hfill \\ \hfill Q_8& =⟨d_{0010},d_{1000},c_{0001},b_{1000},a_{1000}⟩;\hfill \\ \hfill Q_9& =⟨c_{0010}{c}_{0001}-d_{0001}{d}_{0010},a_{0010}{c}_{0001}-b_{0001}{d}_{0010},b_{0001}{c}_{0010}-a_{0010}{d}_{0001},\hfill \\ \hfill & b_{1000}{c}_{0100}-a_{0100}{d}_{1000},a_{1000}{c}_{0100}-b_{0100}{d}_{1000},a_{1000}{a}_{0100}-b_{0100}{b}_{1000}⟩;\hfill \\ \hfill Q_{10}& =⟨d_{0010},d_{1000},c_{0001},c_{0100},a_{1000}{a}_{0100}-b_{0100}{b}_{1000}⟩;\hfill \\ \hfill Q_{11}& =⟨b_{0001},b_{1000},a_{0010},a_{0100},c_{0010}{c}_{0001}-d_{0001}{d}_{0010}⟩.\hfill \end{array}$$

The i-th integrability condition is determined by the variety associated with the ideal $Q_i$. For all but two cases defined by these $Q_i$ we will prove that the corresponding systems are integrable. The dual cases (in the sense defined in the previous section) of the varieties arising from ideals $Q_1-Q_4$ are the varieties of ideals $Q_5-Q_8$, which are written in Appendix A. They do not need to be considered since the proof of integrability for them is exactly the same as in the original cases.

Theorem 4.

For the origin of system (1) to be integrable, it suffices that one of the following is true:

1. 

$d_{0010}=c_{0001}=c_{0100}=b_{1000}=a_{0010}=a_{0100}=0$;

2. 

$d_{0010}=c_{0010}=b_{1000}=a_{0010}=a_{0100}=0$;

3. 

$d_{0010}=d_{1000}=c_{0001}=b_{1000}=a_{0010}=a_{0100}=0$;

4. 

$d_{0010}=c_{0001}=c_{0100}=b_{0100}=a_{0100}=0$;

5. 

$c_{0010}{c}_{0001}-d_{0001}{d}_{0010}=a_{0010}{c}_{0001}-b_{0001}{d}_{0010}=b_{0001}{c}_{0010}-a_{0010}{d}_{0001}=b_{1000}{c}_{0100}-a_{0100}{d}_{1000}=a_{1000}{c}_{0100}-b_{0100}{d}_{1000}=a_{1000}{a}_{0100}-b_{0100}{b}_{1000}=0$,

as well as the dual cases (1*), (2*), (3*), (4*) of (1), (2), (3), (4), respectively, which are written in Appendix A. The systems of cases (1)–(4) and their duals are linearizable.

Proof. 

Case 1. Under these conditions, system (1) becomes:

$$\begin{array}{cc}\hfill \dot{x}& =x(1+a_{1000}x);\hfill \\ \hfill \dot{y}& =y(-1+b_{0100}y+b_{0001}w);\hfill \\ \hfill \dot{z}& =z(i+c_{0010}z);\hfill \\ \hfill \dot{w}& =w(-i+d_{1000}x+d_{0001}w).\hfill \end{array}$$

(27)

The first equation of this system can be transformed to a linear one by the change $X=x{(1+a_{1000}x)}^{-1}$. This gives us $\dot{X}=X$. Then, $x=X{(1-a_{1000}X)}^{-1}$. The fourth equation then becomes:

$$\dot{w}=w(-i+d_{1000}X{(1-a_{1000}X)}^{-1}+d_{0001}w).$$

This equation will be linearized by the substitution $w={\displaystyle \frac{w}{f_1+g_{1}w}}$, as stated in Lemma 1, if we find two functions $f_1$ and $g_1$ such that:

$${\dot{f}}_1=d_{1000}X{(1-a_{1000}X)}^{-1}{f}_1,{\dot{g}}_1-i{g}_1=d_{0001}{f}_1.$$

(28)

The first equation of (28) has the solution $f_1=f_1\left(X\right)={(1-a_{1000}X)}^{-{\displaystyle \frac{d_{1000}}{a_{1000}}}}$. Now, we need to solve for $g_1$, which is a function such that:

$${\dot{g}}_1-i{g}_1=d_{0001}{(1-a_{1000}X)}^{-{\displaystyle \frac{d_{1000}}{a_{1000}}}}.$$

Define $g_1={\sum }_{j\ge 0}{a}_j{X}^j$ and $d_{0001}{(1-a_{1000}X)}^{-{\displaystyle \frac{d_{1000}}{a_{1000}}}}={\sum }_{j\ge 0}{b}_j{X}^j$. Then,

$${\dot{g}}_1-i{g}_1=\sum _{j\ge 0}{a}_j(j-i)X^j=\sum _{j\ge 0}{b}_j{X}^j,$$

which implies that $a_j={\displaystyle \frac{b_j}{j-i}}$. This gives us $\dot{W}=-iW$, where $w={\displaystyle \frac{f_{1}W}{1-g_{1}W}}$. The second equation thus becomes:

$$\dot{y}=y\left(-1+b_{0100}y+b_{0001}{\displaystyle \frac{f_{1}W}{1-g_{1}W}}\right).$$

Similarly, this equation can be linearized using Lemma 1 and the substitution $Y={\displaystyle \frac{y}{f_2+g_{2}y}}$, if we find two functions $f_2$ and $g_2$ such that:

$${\dot{f}}_2=b_{0001}{\displaystyle \frac{f_{1}W}{1-g_{1}W}}{f}_2,{\dot{g}}_2-g_2=b_{0100}{f}_2.$$

(29)

To find a function $f_2$ such that the first equation of (29) is satisfied, we define a function $h_2$ such that $f_2=e^{h_2}$. Then, ${\dot{h}}_2=b_{0001}{\displaystyle \frac{f_{1}W}{1-g_{1}W}}=b_{0001}{f}_{1}W(1+g_{1}W+{\left(g_{1}W\right)}^2+\cdots )=b_{0001}(f_{1}W+f_1{g}_1{W}^2+f_1{g}_1^2{W}^3+f_1{g}_1^3{W}^4+\cdots )=:{\sum }_{j,k\ge 0}{d}_{jk}{X}^j{W}^k$. Set $h_2={\sum }_{j,k\ge 0}{c}_{jk}{X}^j{W}^k$. Then, ${\dot{h}}_2={\sum }_{j,k\ge 0}{c}_{jk}(j-ki)X^j{W}^k$. We can see that $c_{jk}={\displaystyle \frac{d_{jk}}{j-ki}},\forall k\ge 1$ and $\forall j\ge 0$, and $c_{j0}=0,\forall j\ge 0$. We solved for $h_2$ which implies $f_2=e^{h_2}={\sum }_{j\ge 0}{\displaystyle \frac{h_2^j}{j!}}$. Now,

$${\dot{g}}_2-g_2=b_{0100}{f}_2=b_{0100}\left(1+h_2+{\displaystyle \frac{h_2^2}{2!}}+\cdots \right)=:\sum _{j,k\ge 0}{f}_{jk}{X}^j{W}^k.$$

(30)

Set $g_2={\sum }_{j,k\ge 0}{e}_{jk}{X}^j{W}^k$. Then,

$${\dot{g}}_2-g_2=\sum _{j,k\ge 0}{e}_{jk}(j-1-ki)X^j{W}^k.$$

From (30), we can deduce that $e_{jk}={\displaystyle \frac{f_{jk}}{j-1-ki}}$ for $(j,k)\ne (1,0)$. We can infer that the monomial X is absent from the right-hand side of (30), yielding $e_{10}=0$. In fact, $e_{00}=-b_{0100}$ and $e_{j0}=0,\forall j\ge 1$. This gives us a solution for $g_2$, which means that the second equation is also linearized, yielding $\dot{Y}=-Y$, where $y={\displaystyle \frac{f_{2}Y}{1-g_{2}Y}}$. The third equation can be linearized by the substitution $Z=z{(1-i{c}_{0010}z)}^{-1}$, which implies $\dot{Z}=iZ$. Thus, by Theorem 1, system (27) is analytically linearizable and subsequently completely integrable.

Case 2. For this case, system (1) becomes:

$$\begin{array}{cc}\hfill \dot{x}& =x(1+a_{1000}x);\hfill \\ \hfill \dot{y}& =y(-1+b_{0100}y+b_{0001}w);\hfill \\ \hfill \dot{z}& =z(i+c_{0100}y+c_{0001}w);\hfill \\ \hfill \dot{w}& =w(-i+d_{1000}x+d_{0001}w).\hfill \end{array}$$

(31)

We can see that the system of the first, second and fourth equation is independent of z and is the same system as in Case 1. This means that it can be linearized by the same procedure, giving us the same expressions for $f_1,f_2,g_1,g_2$ and $h_2$. So far, we have $\dot{X}=X,\dot{Y}=-Y,\dot{W}=-iW$. The expression of the third equation is then:

$$\dot{z}=z\left(i+c_{0001}{\displaystyle \frac{f_{1}W}{1-g_{1}W}}+c_{0100}{\displaystyle \frac{f_{2}Y}{1-g_{2}Y}}\right).$$

This equation can be linearized by Lemma 2 and the substitution $Z=z{e}^{f_3+f_4}$, if we find functions $f_3$ and $f_4$ such that ${\dot{f}}_3=-c_{0001}{\displaystyle \frac{f_{1}W}{1-g_{1}W}}$ and ${\dot{f}}_4=-c_{0100}{\displaystyle \frac{f_{2}Y}{1-g_{2}Y}}$. One can easily see that $f_3=-{\displaystyle \frac{c_{0001}}{b_{0001}}}{h}_2$ is a solution to the first equation. For the second equation, we write:

$${\dot{f}}_4=-c_{0100}{\displaystyle \frac{f_{2}Y}{1-g_{2}Y}}=-c_{0100}{f}_{2}Y(1+g_{2}Y+{\left(g_{2}Y\right)}^2+\cdots )=:\sum _{j,k,l\ge 0}{n}_{jkl}{X}^j{Y}^k{W}^l.$$

(32)

Set $f_4={\sum }_{j,k,l\ge 0}{m}_{jkl}{X}^j{Y}^k{W}^l$. Then, from (32), we have:

$${\dot{f}}_4=\sum _{j,k,l\ge 0}{m}_{jkl}(j-k-l·i)X^j{Y}^k{W}^l=\sum _{j,k,l\ge 0}{n}_{jkl}{X}^j{Y}^k{W}^l.$$

One finds that there are no monomials of the form ${\left(XY\right)}^j$ in the expression because the product $f_{2}Y=Y+h_{2}Y+{\displaystyle \frac{h_2^2}{2!}}Y+\cdots$ contains only Y and monomials whose degree of W is greater than zero as a consequence of the definition of $h_2$. This implies that we can set $m_{jj0}=0$ for all $j\ge 0$ and $m_{jkl}={\displaystyle \frac{n_{jkl}}{j-k-l·i}}$ for any other triple $(j,k,l)$ of non-negative integers. This gives an expression for $f_4$ which implies $\dot{Z}=iZ$. System (31) is linearized and, as a result of that, integrable.

Case 3. Under this assumption, system (1) reduces to the following form:

$$\begin{array}{cc}\hfill \dot{x}& =x(1+a_{1000}x);\hfill \\ \hfill \dot{y}& =y(-1+b_{0100}y+b_{0001}w);\hfill \\ \hfill \dot{z}& =z(i+c_{0100}y+c_{0010}z);\hfill \\ \hfill \dot{w}& =w(-i+d_{0001}w).\hfill \end{array}$$

(33)

It is clear that the system of the first, second and third equation is the same as the system of the first, second and fourth equation of the first case for $d_{1000}=0$. Consequently, the linearization follows the same procedure as in Case 1, employing identical functions under the constraint $d_{1000}=0$. The third equation then becomes:

$$\dot{z}=z\left(i+c_{0100}{\displaystyle \frac{Y{f}_1}{1-g_{1}Y}}+c_{0010}z\right),$$

where $f_1={(1-i{d}_{0001}W)}^{-{\displaystyle \frac{b_{0001}}{d_{0001}}}}$ and $g_1={\sum }_{j\ge 0}{\displaystyle \frac{a_j}{j·i-1}}{W}^j$, where $a_j$ is the coefficient of $W^j$ in the expansion of $b_{0100}{(1-i{d}_{0001}W)}^{-{\displaystyle \frac{b_{0001}}{d_{0001}}}}$. This equation can be linearized using Lemma 1 if we find functions $f_2$ and $g_2$ such that:

$${\dot{f}}_2=c_{0100}{\displaystyle \frac{Y{f}_1}{1-g_{1}Y}}{f}_2,{\dot{g}}_2+i{g}_2=c_{0010}{f}_2.$$

(34)

To find a function $f_2$ such that the first equation of (34) is satisfied, we define a function $h_2$ such that $f_2=e^{h_2}$. Then, ${\dot{h}}_2=c_{0100}{\displaystyle \frac{Y{f}_1}{1-g_{1}Y}}=c_{0100}(f_{1}Y+f_1{g}_1{Y}^2+f_1{g}_1^2{Y}^3+\cdots )=:{\sum }_{j,k\ge 0}{d}_{jk}{Y}^j{W}^k$. Set $h_2={\sum }_{j,k\ge 0}{c}_{jk}{Y}^j{W}^k$. Then, ${\dot{h}}_2={\sum }_{j,k\ge 0}{c}_{jk}(-j-k·i)Y^j{W}^k$. We can see that $c_{jk}=-{\displaystyle \frac{d_{jk}}{j+k·i}},\forall j\ge 1$ and $\forall k\ge 0$. Also, $c_{0k}=0,\forall k\ge 0$. This gives us solutions for $f_2$ and $g_2$ and so the third equation is linearized, giving $\dot{Z}=iZ$. The system is linearized, which means that is integrable.

Case 4. System (1) with these conditions is:

$$\begin{array}{cc}\hfill \dot{x}& =x(1+a_{1000}x+a_{0010}z);\hfill \\ \hfill \dot{y}& =y(-1+b_{1000}x+b_{0001}w);\hfill \\ \hfill \dot{z}& =z(i+c_{0010}z);\hfill \\ \hfill \dot{w}& =w(-i+d_{1000}x+d_{0001}w).\hfill \end{array}$$

(35)

The third equation is linearized via the substitution $Z=z{(1-i{c}_{0010}z)}^{-1}$. This implies that $\dot{Z}=iZ$. Then, $z=Z{(1+i{c}_{0010}Z)}^{-1}$. The first equation then becomes:

$$\dot{x}=x(1+a_{1000}x+a_{0010}Z{(1+i{c}_{0010}Z)}^{-1}).$$

As stated in Lemma 1, this equation can be linearized if we find two functions $f_1$ and $g_1$ such that:

$${\dot{f}}_1=a_{0010}Z{(1+i{c}_{0010}Z)}^{-1}{f}_1,{\dot{g}}_1+g_1=a_{1000}{f}_1.$$

(36)

The solution to the first equation of (36) is $f_1=f_1\left(Z\right)={(1+i{c}_{0010}Z)}^{-{\displaystyle \frac{a_{0010}}{c_{0010}}}}$. Now, we need to solve for $g_1$, which is a function such that:

$${\dot{g}}_1+g_1=a_{1000}{(1+i{c}_{0010}Z)}^{-{\displaystyle \frac{a_{0010}}{c_{0010}}}}.$$

Define $g_1={\sum }_{j\ge 0}{a}_j{Z}^j$ and $a_{1000}{(1+i{c}_{0010}Z)}^{-{\displaystyle \frac{a_{0010}}{c_{0010}}}}={\sum }_{j\ge 0}{b}_j{Z}^j$. Then:

$${\dot{g}}_1+g_1=\sum _{j\ge 0}{a}_j(j·i+1)Z^j=\sum _{j\ge 0}{b}_j{Z}^j,$$

which implies that $a_j={\displaystyle \frac{b_j}{j·i+1}}$ and a solution for $g_1$. This gives us $\dot{X}=X$, where $x={\displaystyle \frac{f_{1}X}{1-g_{1}X}}$. The fourth equation then becomes:

$$\dot{w}=w\left(-i+d_{1000}{\displaystyle \frac{f_{1}X}{1-g_{1}X}}+d_{0001}w\right).$$

This equation can also be linearized by Lemma 1 and the substitution $W={\displaystyle \frac{w}{f_2+g_{2}w}}$ if we find two functions $f_2$ and $g_2$ such that:

$${\dot{f}}_2=d_{1000}{\displaystyle \frac{f_{1}X}{1-g_{1}X}}{f}_2,{\dot{g}}_2-i{g}_2=d_{0001}{f}_2.$$

(37)

To find a function $f_2$ such that the first equation of (37) holds, we define a function $h_2$ such that $f_2=e^{h_2}$. Then, ${\dot{h}}_2=d_{1000}{\displaystyle \frac{f_{1}X}{1-g_{1}X}}=d_{1000}{f}_{1}X(1+g_{1}X+{\left(g_{1}X\right)}^2+\cdots )=d_{1000}(f_{1}X+f_1{g}_1{X}^2+f_1{g}_1^2{X}^3+f_1{g}_1^3{X}^4+\cdots )=:{\sum }_{j,k\ge 0}{d}_{jk}{X}^j{Z}^k$. Set $h_2={\sum }_{j,k\ge 0}{c}_{jk}{X}^j{Z}^k$. Then, ${\dot{h}}_2={\sum }_{j,k\ge 0}{c}_{jk}(j+k·i)X^j{Z}^k$. Therefore, $c_{jk}={\displaystyle \frac{d_{jk}}{j+k·i}},\forall k,j\ge 1$, and $c_{0j}=0,\forall j\ge 0$. Thus, we solved for $h_2$ and $f_2=e^{h_2}={\sum }_{j\ge 0}{\displaystyle \frac{h_2^j}{j!}}$. Now,

$${\dot{g}}_2-i{g}_2=d_{0001}\left(1+h_2+{\displaystyle \frac{h_2^2}{2!}}+\cdots \right)=:\sum _{j,k\ge 0}{f}_{jk}{X}^j{Z}^k.$$

(38)

Set $g_2={\sum }_{j,k\ge 0}{e}_{jk}{X}^j{Z}^k$. Then,

$${\dot{g}}_2-i{g}_2=\sum _{j,k\ge 0}{e}_{jk}(j+(k-1)·i)X^j{Z}^k.$$

From (38), it is clear that $e_{jk}={\displaystyle \frac{f_{jk}}{j+(k-1)·i}}$ for $(j,k)\ne (0,1)$. The monomial Z is not present in the right-hand side of (38), yielding $e_{01}=0$. In fact, $e_{00}=i{d}_{0001}$ and $e_{0j}=0,\forall j\ge 2$. This gives a solution for $g_2$, which means that the fourth equation is also linearized, that is $\dot{W}=-iW$, where $w={\displaystyle \frac{f_{2}W}{1-g_{2}W}}$. The expression of the second equation is then:

$$\dot{y}=y\left(-1+b_{1000}{\displaystyle \frac{f_{1}X}{1-g_{1}X}}+b_{0001}{\displaystyle \frac{f_{2}W}{1-g_{2}W}}\right).$$

This equation can be brought to a linear one by Lemma (2) and the change $Y=y{e}^{f_3+f_4}$ if we find functions $f_3$ and $f_4$ such that ${\dot{f}}_3=-b_{1000}{\displaystyle \frac{f_{1}X}{1-g_{1}X}}$ and ${\dot{f}}_4=-b_{0001}{\displaystyle \frac{f_{2}W}{1-g_{2}W}}$. It is obvious that $f_3=-{\displaystyle \frac{b_{1000}}{d_{1000}}}{h}_2$ is a solution to the first equation. For the second equation, we write:

$${\dot{f}}_4=-b_{0001}{\displaystyle \frac{f_{2}W}{1-g_{2}W}}=-b_{0001}{f}_{2}W(1+g_{2}W+{\left(g_{2}W\right)}^2+\cdots )=:\sum _{j,k,l\ge 0}{n}_{jkl}{X}^j{Z}^k{W}^l.$$

(39)

Set $f_4={\sum }_{j,k,l\ge 0}{m}_{jkl}{X}^j{Z}^k{W}^l$. Then, from (39), we have:

$${\dot{f}}_4=\sum _{j,k,l}{m}_{jkl}(j+(k-l)·i)X^j{Y}^k{Z}^l=\sum _{j,k,l\ge 0}{n}_{jkl}{X}^j{Z}^k{W}^l.$$

We can see that there are no monomials of the form ${\left(ZW\right)}^k$ in the expression ${\sum }_{j,k,l\ge 0}{n}_{jkl}{X}^j{Z}^k{W}^l$ because the product $f_{2}W=W+h_{2}W+{\displaystyle \frac{h_2^2}{2!}}W+\cdots$ contains only W and monomials whose degree of X is greater than zero as a consequence of the definition of $f_2$. This implies that we can set $m_{0jj}=0,\forall j\ge 0$ and $m_{jkl}={\displaystyle \frac{n_{jkl}}{j+(k-l)·i}}$ for any other triple $(j,k,l)$ of non-negative integers. This gives a solution for $f_4$ and implies $\dot{Y}=-Y$. System (35) is linearized and consequently integrable.

Case 5. For system (1) the ideal (25) is:

$$\begin{array}{c}J=⟨a_{1000}+{\sigma }_1{b}_{0100},a_{0100}+{\sigma }_2{b}_{1000},a_{0010}+{\sigma }_3{b}_{0001},c_{0100}+{\sigma }_2{d}_{1000},\hfill \\ \hfill c_{0001}+{\sigma }_4{d}_{0010},c_{0010}+{\sigma }_3{d}_{0001},{\sigma }_1{\sigma }_2-1,{\sigma }_3{\sigma }_4-1⟩.\end{array}$$

Using the elimination theorem, we computed $\mathcal{I}=J\cap \mathbb{C}[a,b,c,d]$ and obtained that $\mathcal{I}=Q_9$. Thus, by Theorem 3, the systems whose parameters belong to $\mathbf{V}\left(Q_9\right)$ are integrable. □

For systems from the components $Q_{10}$ and $Q_{11}$ the existence of one first integral can be established easily. Namely, systems from $\mathbf{V}\left(Q_{10}\right)$ admit a first integral $\phi =zw{\left(1-i{c}_{0010}z\right)}^{-1}{\left(1+i{d}_{0001}w\right)}^{-1}$ and systems from $Q_{11}$ have a first integral $\phi =xy{(1+a_{1000}x)}^{-1}{(1-b_{0100}y)}^{-1}$. Additionally, a calculation of the normal form up to order nine reveals that systems of these cases are not linearizable. The inability to find the second independent first integral prevents the complete analysis of integrability, but motivates the following conjecture.

Conjecture 1.

Systems satisfying the condition: $d_{0010}=d_{1000}=c_{0001}=c_{0100}=a_{1000}{a}_{0100}-b_{0100}{b}_{1000}=0$ or $b_{0001}=b_{1000}=a_{0010},a_{0100}=c_{0010}{c}_{0001}-d_{0001}{d}_{0010}=0$ are integrable.

It appears new approaches have to be developed in order to prove complete integrability for the cases mentioned in the conjecture.

6. Conclusions

In this paper, we examined the integrability and linearizability properties of the family of four-dimensional Lotka–Volterra systems given by (1). Through the development and application of a systematic methodology, we established explicit parameter conditions that guarantee the existence of two independent analytic first integrals.

Furthermore, we identified that within the set of integrable systems, there exist time-reversible systems. By exploiting the symmetry properties of these systems, we proved their integrability and demonstrated the connection between time-reversal symmetry and the existence of analytic first integrals.

Overall, the results in this paper contribute to a deeper understanding of the structure and dynamics of four-dimensional Lotka–Volterra systems and provide a general framework for analyzing integrability in higher-dimensional polynomial systems. Future work may focus on extending this methodology to more general classes of polynomial vector fields and investigating global dynamical properties of the integrable and reversible cases, as well as studying perturbations of such systems using the Andronov-Melnikov approach in the spirit of [47,48].

Another important direction for further research is the study of small perturbations of the integrable systems considered here and similar ones using averaging methods and normal form techniques described in [49]. This includes the analysis of the persistence and bifurcation of invariant tori in the spirit of KAM theory. Although KAM theory mainly deals with Hamiltonian systems, as is shown in [50,51], similar phenomena are observed also in non-Hamiltonian systems. Starting from the obtained integrable cases, one can analyze perturbations preserving the (1:−1:i:−i) resonance and derive averaged systems governing the slow dynamics near invariant tori, and then investigate the persistence of quasi-periodic motions and look for conditions ensuring the survival of invariant tori in the sense of KAM theory.