1 Introduction↩︎

A large part of Jarmo Hietarinta’s academic career he has spent searching for integrable systems [1]–[8], and one of his early interests has been Liouville integrability [9]. In this paper, we search for Liouville integrable Lotka-Volterra (LV) systems, i.e. ordinary differential equations of the form \[\label{LV} \dot{x}_i=x_i\left(r_i+\sum_{j=1}^n A_{ij}x_j\right),\quad i=1,2,\ldots,n,\tag{1}\] where the vector \(\mathbf{r}\) and matrix \(\mathbf{A}\) do not depend on \(\mathbf{x},t\), and \(\dot{\mathbf{x}}\) denotes the \(t\)-derivative of \(\mathbf{x}\). They are a generalisation of the 2-dimensional system, derived independently by Lotka and by Volterra [10], [11]. So, in essence, we will be extending integrable Hamiltonian LV systems from 2 to \(n\) dimensions, cf. [12].

LV systems have been studied extensively and they found applications in e.g. biology and chemistry [13]–[19]. One of the interesting things about LV systems is that they can exhibit regular or chaotic behaviour, or both. Of course, systems with maximum regularity are integrable.

A Darboux Polynomial (DP) is a polynomial \(P\) whose derivative, \(\dot{P}\), is proportional to \(P\). The entire \(n(n+1)\)-parameter class of LV-systems 1 admits \(n\) such polynomials, namely the variables \(x_i\), \(i=1,\ldots,n\). By specialisation of the parameters, we obtain subclasses with additional DPs, which give rise to integrals, or other mathematical structures such as a Poisson bracket. Recently, the authors have established large classes of integrable cases. This includes:

• the study [20], of inhomogeneous LV systems of the form \[\label{LVF} \dot{x}_i=x_i\,\big(r_i + \sum_{j>i} x_j - \sum_{j<i}x_j\big),\qquad r_i=\begin{cases} b, & i=1,\ldots,k,\\ c, & i=k+1,\ldots, k+l,\\ d, & i=k+l+1,\ldots, k+l+m=n, \end{cases}\tag{2}\] where \(c=b+d\) and \(km\neq 0\). The 2-parameter class 2 contains Liouville integrable, Liouville superintegrable, nonholonomically integrable, and ‘integrable-or-not’ subclasses. For the latter we were not able to find a Hamiltonian structure or more than \(n-3\) integrals. Still the entire class was shown to be solvable, using the solutions to contracted 2- and 3-component systems given in terms of the Lambert W function.

• the construction of LV tree-systems in [21], [22]. These are homogeneous superintegrable \((3n-2)\)-parameter classes of \(n\)-component LV systems. They admit \(n-1\) functionally independent integrals that involve \(n-1\) linear DPs in two variables which correspond to the edges of a tree.

• the classification of classes of \(n\)-component LV systems with linear DPs in any number of variables, for \(n<6\), up to linear transformations. These systems are associated to (LV admissible) hypergraphs. One new superintegrable \(3n-2=13\) parameter family of \(n=5\) component LV systems was found, not equivalent to a tree system.

In this paper, we assume that \(\mathbf{A}\) is an anti-symmetric constant \(n \times n\) matrix, starting with \(n=2m\). In the homogeneous case, this implies that the LV-system is a Hamiltonian system, with Poisson bracket \[\label{PBA} \{x_i,x_j\}=A_{i,j}x_ix_j\tag{3}\] and Hamiltonian \(H=\sum_{i=1}^n x_i\). In other words, the equation \[\label{HF} \dot{x}_i=\{x_i,H\},\quad i=1,\ldots,n,\tag{4}\] equals 1 with \(\mathbf{r}=0\). We follow a similar approach to the one taken in [21]–[23], i.e., we require that certain linear functions are DPs of the system (each of them giving rise to an integral). Note that we do not need as many as before. Whereas for superintegrability \(n-1\) were needed, here, besides the Hamiltonian, \(n/2-1\) DPs suffice. On the other hand, we do require the corresponding integrals to Poisson commute. Thus we establish Liouville integrability which is defined as follows.

We present a large number of homogeneous and inhomogeneous Liouville integrable multi-parameter families of \(n\)-dimensional Lotka-Volterra systems, for each \(n\in\mathbf{N}\). In Section 2.1, we present \(\frac{m^{2}}{4}+\frac{m}{2}+\frac{1-\left(-1\right)^{m}}{8}\) homogeneous \(n=2m\) dimensional \(3m-2\) parameter families of LV systems, which have \(m\) integrals in involution. In Section 2.2, we present equally many odd \(n=2m-1\) dimensional LV systems, for which we have \(m-1\) integrals in involution and 1 Casimir. In Section 3, we show that each even-dimensional system has a subsystem to which one can add a inhomogeneous term of the form \(\mathbf{r}=(r,\ldots,r,s)\). In odd dimensions, this is not the case. In Section 4, we provide explicit details for the families of Liouville integrable LV systems we have found in dimensions 3,4,5,6, as well as for some Liouville integrable LV systems of different type in dimension \(n=10\).

The number of different Liouville integrable classes of LV systems we provide in this paper grows quadratically, whereas the number of superintegrable classes of tree-systems grows exponentially with \(n\). The number of parameters in each class, \(3n/2-2\), is about half the number of parameters for tree-systems. We do not claim to have caught all the Liouville integrable fish in the LV pond, however, it seems that the cases that are not of ‘type [\(j,k,l\)]’, have more than the required number of DPs/integrals and less than \(3n/2-2\) parameters.

2 Homogeneous Liouville integrable LV systems↩︎

In this section, we consider homogeneous \(n\)-dimensional Hamiltonian systems 4 , with anti-symmetric matrix \(\mathbf{A}\) and Hamiltonian \[\label{H} H=x_1+x_2+\cdots+x_n.\tag{5}\] Integrals for 4 are functions \(F\) for which \(\dot{F}=\{F,H\}=0\). A Darboux Polynomial (DP) is a polynomial \(P\) for which \[\dot{P}=\{P,H\}=CP,\] in which case \(C\) is called the cofactor of \(P\), also denoted \(C[P]\). Clearly, the Hamiltonian itself is an integral, i.e., a DP with cofactor 0. More generally, a function \(P\) which satisfies \(\dot{P}=\{P,H\}=CP\) is called a second integral [13]. A product of powers of DPs, \[\Pi=\prod_i P_i^{\alpha_i}\] is a second integral with cofactor \(C[\Pi]=\sum_i\alpha_iC[P_i]\).

We will require linear combinations of the form \[\label{DP} P_I=\alpha_1 x_{i_1} + \alpha_2 x_{i_2} + \cdots + \alpha_k x_{i_k},\tag{6}\] where \(I=\{i_1,\ldots,i_k\}\subset\mathbb{N}_n=\{x_1,\ldots,x_n\}\), to be DPs. This imposes conditions on the matrix \(\mathbf{A}\), as given in [21] for \(k=2\), or in [23] for \(k>2\). Due to the antisymmetry of \(\mathbf{A}\), the coefficients \(\alpha_i\) in \(P_I\) are all equal; we take \(\alpha_i=1\) for all \(i\). If the matrix \(\mathbf{A}\) has full rank, then each DP gives rise to a integral. Indeed, if \(\mathbf{B}\) is the \(1\times n\) vector such that \(C[P_I]=\sum_{i=1}^n B_ix_i\), and \[\mathbf{Z}=-\mathbf{B}.\mathbf{A}^{-1},\] then \[\label{integral} P_I\prod_{i=1}^n x_i^{Z_{i}}\tag{7}\] is an integral, cf. [21]. Further conditions on the matrix \(\mathbf{A}\) are imposed by requiring the integrals to Poisson commute. A homogeneous \(n\)-dimensional Hamiltonian LV system with \(k\) integrals of the form 7 which are pairwise commuting will be represented by a hypergraph of order \(n\) and size \(k\) (i.e. a subset, with \(k\) elements, of the powerset of \(n\) symbols).

For example, the Hamiltonian LV system with matrix \[\label{fa} \mathbf{A}=\begin{pmatrix} 0 & a_{1} & b_{1} & b_{1} & b_{1} & c_{1} \\ -a_{1} & 0 & b_{1} & b_{1} & b_{1} & c_{1} \\ -b_{1} & -b_{1} & 0 & a_{2} & a_{3} & a_{5} \\ -b_{1} & -b_{1} & -a_{2} & 0 & a_{4} & a_{5} \\ -b_{1} & -b_{1} & -a_{3} & -a_{4} & 0 & a_{5} \\ -c_{1} & -c_{1} & -a_{5} & -a_{5} & -a_{5} & 0 \end{pmatrix}\tag{8}\] has DPs \(x_1+x_2\) and \(x_3+x_4+x_5\), with corresponding integrals \[\left(x_{1}+x_{2}\right) \left( \frac{x_{4}^{a_{3}}}{x_{3}^{a_{4}}x_{5}^{a_{2}}} \right)^{\frac{c_{1}}{a_{5} \left(a_{2}-a_{3}+a_{4}\right)}} x_{6}^{\frac{b_{1}}{a_{5}}},\quad \left(x_{3}+x_{4}+x_{5}\right) \left( \frac{x_{4}^{a_{3}}}{x_{3}^{a_{4}}x_{5}^{a_{2}}} \right)^{\frac{1}{a_{2}-a_{3}+a_{4}}},\] that Poisson commute. It is represented by the hypergraph in Figure [Fe].

We note that this representation differs from the one in [23], in more than one way. First of all, the matrices in [23] are not, in general, antisymmetric. In the current paper, the matrix \(\mathbf{A}\) is assumed to be anti-symmetric which implies that, in all cases, the function \(H\) is a DP (and an integral), but we omit the corresponding hyperedge (involving all vertices). Secondly, as stated above, here we require the specified integrals, obtained from 7 , to commute.

In Section 2.1, we consider even-dimensional LV systems. In each case, the matrix \(\mathbf{A}\) will have full rank, so integrals can be calculated using the formula 7 . We do however present different integrals, that are (or can be made) homogeneous of weight zero. This allows us to extend the systems to the inhomogeneous case, which we do in Section 3. In Section 2.2 we show that each even-dimensional system can be reduced to an odd-dimensional system, whilst retaining its Liouville integrability. In the \(n=2m\) dimensional case, we have \(m\) functionally independent integrals (including the Hamiltonian) in involution. In the \(n=2m-1\) dimensional case, there are \(m-1\) functionally independent integrals (including the Hamiltonian) in involution, as well as a Casimir.

2.1 Even-dimensional homogeneous Liouville integrable LV systems↩︎

We will show there exists a \(2m\)-dimensional homogeneous Liouville integrable LV system with \(3m-2\) parameters associated to each hypergraph of order \(2m\) displayed in Fig. [F0]. These hypergraphs will be labeled by the number of degree 2 edges and the two (ordered) numbers of nested edges. For example, the hypergraph in Fig. [Fe] is labeled by \([1,1,0]\), and the hypergraphs displayed explicitly in Fig. [F0] are labeled by \([m-1,0,0]\), \([m-2,1,0]\), \([m-3,2,0]\), \([m-k-1,k,0]\), \([m-k-2,k,1]\), and \([m-k-l-1,k,l]\) respectively (with \(k\geq l\geq 0\) and \(m>k+l\)).

2.1.1 The even-dimensional homogeneous type \([j,0,0]\) LV system↩︎

The type \([j,0,0]\) LV system corresponds to the graph in Fig. [F1].

We denote by \(\mathbf{1}^{i,k}\) the \(i\times k\) matrix with entries \(1\). Using \(2\times 2\) matrices \[\label{JKLehj00} \mathbf{J}_i=\begin{pmatrix} 0 & a_i \\ -a_i & 0 \end{pmatrix},\; \mathbf{K}^i_k=\mathbf{K}^i_k(h)=\frac{b_ic_k-b_kc_i}{a_{h}}\mathbf{1}^{2,2},\tag{9}\] we define the \(2j\times2j\) matrix \[\label{Mjh} \mathbf{M}^j_h=\begin{pmatrix} \mathbf{J}_1 & \mathbf{K}^1_2 & \mathbf{K}^1_3 & \cdots & \mathbf{K}^1_j\\ -\mathbf{K}^1_2 & \mathbf{J}_2 & \mathbf{K}^2_3 & \cdots & \mathbf{K}^2_j\\ -\mathbf{K}^1_3 & - \mathbf{K}^2_3 & \mathbf{J}_3 & \cdots & \mathbf{K}^3_j\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ -\mathbf{K}^1_j & -\mathbf{K}^2_j & -\mathbf{K}^3_j & \cdots & \mathbf{J}_{j} \end{pmatrix},\tag{10}\] and we define the \(2j\times2\) matrix \[\label{Lj} \mathbf{L}_j=\begin{pmatrix} b_1\mathbf{1}^{2,1} & c_1\mathbf{1}^{2,1} \\ b_2\mathbf{1}^{2,1} & c_2\mathbf{1}^{2,1} \\ \vdots & \vdots \\ b_j\mathbf{1}^{2,1} & c_j\mathbf{1}^{2,1} \end{pmatrix}.\tag{11}\] The transpose of a matrix will be indicated by a superscript \(^T\).

There are \(3j+1\) parameters in the matrix ?? .

2.1.2 The even-dimensional homogeneous type \([0,k,0]\) LV system↩︎

The type \([0,k,0]\) LV system corresponds to the hypergraph in Fig. [F2].

We define an antisymmetric \(n\times n\) matrix \(\mathbf{N}^n\) by, with \(k<l\), \[\label{DN} N^n_{k,l}= \begin{cases} a_{3j-1} & l=2j \\ a_{3j-2} & k\leq 2j, l=2j+1 \\ a_{3j} & k= 2j, l=2j+1, \end{cases}\tag{18}\] and \(N^n_{l,k}=-N^n_{k,l}\). We denote \(P_{a,b}=x_a+x_{a+1}+\cdots x_b\), following notation introduced in [20].

2.1.3 The even-dimensional homogeneous type \([0,k,l]\) LV system↩︎

The type \([0,k,l]\) LV system corresponds to the hypergraph in Fig. [F3].

Let \(\sigma_z\) be the shift operator \(\sigma_z(z_i)=z_{i+1}\). We denote \(\mathbf{N}^m_h=\sigma_a^{h-1}\mathbf{N}^m\), and \[\label{gki} G^k_i=\sigma_x^{2k+1}\sigma_a^{3k+1}G_i.\tag{28}\]

2.1.4 The even-dimensional homogeneous type \([j,k,l]\) LV system↩︎

The type \([j,k,l]\) LV system is the most general system we consider, it corresponds to the last hypergraph in Fig. [F0], and it includes the previous cases. We adapt the formulas 11 , ?? , and 28 : \[\label{Ljkl} \mathbf{L}_{j,k,l}=\begin{pmatrix} b_1\mathbf{1}^{2,2k+1} & c_1\mathbf{1}^{2,2l+1} \\ b_2\mathbf{1}^{2,2k+1} & c_2\mathbf{1}^{2,2l+1} \\ \vdots & \vdots \\ b_j\mathbf{1}^{2,2k+1} & c_j\mathbf{1}^{2,2l+1} \end{pmatrix},\tag{29}\] \[\label{Fb} \overline{F}_i=P_{2i-1,2i}^{a_{j+3k+1}}P_{2j+1,2(j+k)+1}^{-c_{i}}P_{2(j+k+1),n}^{b_{i}},\tag{30}\] with \(n=2(j+k+l+1)\), and \[\label{gjki} G^{j}_i=\sigma_x^{2j}\sigma_a^{j}G_i,\quad G^{j,k}_i=\sigma_x^{2(j+k)+1}\sigma_a^{j+3k+1}G_i.\tag{31}\] We denote the \(2(k+l+1)\times 2(k+l+1)\) matrix ?? by \(\mathbf{A}^{k,l}\) , and define \(\mathbf{A}^{k,l}_h=\sigma_a^{h-1}\mathbf{A}^{k,l}\). We also use definition 10 in the following proposition.

2.2 Odd-dimensional homogeneous Liouville integrable LV systems↩︎

If the matrix \(\mathbf{A}\) does not have full rank, Casimirs can be found by determining the null-space of \(\mathbf{A}\) (which, as \(\mathbf{A}\) is antisymmetric, equals the null-space of its transpose). Indeed, if \(\sum_j A_{i,j}q_j=0\) for all \(i\), and \(\mathcal{C}=\prod_h x_h^{q_h}\), then \[\{x_i,\mathcal{C}\} = x_i \mathcal{C}\left(\sum_{j} A_{i,j} q_j \right)=0.\] We will obtain odd \(n=2m-1\) dimensional systems from even \(n+1=2m\) dimensional systems by reduction, i.e. setting \(x_{2m}=0\). Integrals of the \(2m\)-dimensional system which do not depend on \(x_{2m}\), will be integrals of the reduced system. The reduced system has one less integral, but it makes up for that by acquiring a Casimir.

2.2.1 The odd-dimensional homogeneous \([j,0,0]\) LV system↩︎

The matrix of the \(n=2j+1\) dimensional LV system of type \([j,0,0]\) is obtained from the matrix of the type \([j,0,0]\) system by deleting the last row and last column. It has \(3j+1\) parameters (the same number as the \(2j+2\)-dimensional type \([j,0,0]\) system). Integrals are obtained from ?? ; we define \[\begin{align} Q_i&=\left(\frac{F_i^{b_{i+1}}}{F_{i+1}^{b_i}}\right)^{1/a_{j+1}} \notag \\ &=\frac{\left(x_{2 i -1}+x_{2 i}\right)^{b_{i +1}} }{\left(x_{2 i +1}+x_{2 i +2}\right)^{b_{i}}}x_{n}^{a_{i,i+1}}, \label{Qi} \end{align}\tag{33}\] where \(a_{i,i+1}\) is defined in 12 .

2.2.2 The odd-dimensional homogeneous \([0,k,0]\) LV system↩︎

2.2.3 The odd-dimensional homogeneous \([0,k,l]\) LV system↩︎

The matrix \(\mathbf{A}\) of the \(n=2(k+l)+1\) dimensional LV system of type \([0,k,l]\) is obtained from the matrix of the type \([0,k,l]\) system by deleting the last row and last column. It has \(3(k+l)-1\) parameters.

2.2.4 The odd-dimensional homogeneous \([j,k,l]\) LV system↩︎

The matrix of the \(n=2(j+k+l)+1\) dimensional LV system of type \([j,k,l]\) is obtained from the matrix of the even-dimensional type \([j,k,l]\) system, by deleting the last row and last column. It has \(3(j+k+l)-1\) parameters. We define \[\begin{align} \overline{Q}_i&=\left(\frac{\overline{F}_i^{b_{i+1}}}{\overline{F}_{i+1}^{b_i}}\right)^{1/a_{j+3k+1}} \notag \\ &=\frac{P_{2 i -1,2 i}^{b_{i +1}} }{P_{2 i +1,2 i +2}^{b_{i}}}P_{2j+1,2(j+k)+1}^{a_{i,i+1}}, \label{oQi} \end{align}\tag{36}\] where \(a_{i,i+1}\) is defined in 32 .

We note that the type \([j,k,0]\) system arises as a special case of the above.

3 Nonhomogeneous Liouville integrable LV systems↩︎

The family of Lotka-Volterra equations 1 is Hamiltonian, or Poisson, if \(\mathbf{A}\) is skew and \(\mathbf{r}\in\text{Im}(\mathbf{A})\). In terms of the Poisson bracket 3 , the system (1 ) can be written as \(\dot{\mathbf{x}}=\{\mathbf{x},H\}\), where the Hamiltonian is given by \[\label{hamLV} H = \sum_{i=1}^{n} (x_i + k_i \ln x_i), \qquad \text{ with } \mathbf{A}\mathbf{k}=\mathbf{r}.\tag{38}\]

A function \(P(\mathbf{x})\) is homogeneous of weight \(d\) if \(P(\lambda\mathbf{x})=\lambda^dP(\mathbf{x})\). The following theorem is an improved version of [24]. Let \(I=\{i_1,\ldots,i_k\}\subset \mathbb{N}_n\), and denote \(\mathbf{x}_I=(x_{i_1},\ldots,x_{i_k})\).

It follows that if a homogeneous LV system admits an integral \(F\) which is homogeneous of weight 0, then the inhomogeneous LV system obtained by adding \(\mathbf{r}=r\mathbf{x}\) to the right-hand-side admits the same integral. Note that, in Theorem 1, the only relevant equations are the ones for \(\mathbf{x}_I\), the equations for the complementary variables \(\mathbf{x}_{I^c}\) can be altered differently.

3.1 Even-dimensional inhomogeneous Liouville integrable LV systems↩︎

3.1.1 The even-dimensional inhomogeneous \([j,0,0]\) LV system↩︎

The \(n=2(j+1)\) dimensional inhomogeneous LV system of type \([j,0,0]\), of the form 1 with constant \(r_i=r\), has matrix \[\label{mnhj00ei} \mathbf{A}=\begin{pmatrix} J_1 & K_{1,2} & K_{1,3} & \cdots & K_{1,j} & L_1 \\ -K_{1,2} & J_2 & K_{2,3} & \cdots & K_{2,j} & L_2 \\ -K_{1,3} & - K_{2,3} & J_3 & \cdots & K_{3,j} & L_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ -K_{1,j} & -K_{2,j} & -K_{3,j} & \cdots & J_{j} & L_{j} \\ -L_1^T & -L_2^T & -L_3^T & \cdots & -L_{j}^T & J_{j+1} \end{pmatrix}\tag{39}\] where \[\label{JKLeij00} J_i=\begin{pmatrix} 0 & a_i \\ -a_i & 0 \end{pmatrix},\; K_{i,k}=(c_i-c_k)\mathbf{1}^{2,2},\; L_i=\begin{pmatrix} c_i - a_{j+1} & c_i \\ c_i - a_{j+1} & c_i \end{pmatrix}.\tag{40}\]

3.1.2 The even-dimensional inhomogeneous type \([j,k,l]\) LV system↩︎

The integrals 31 are homogeneous of weight 0. The only adjustment to be made to the \([j,k,l]\) LV system is the substitution \(b_i=c_i-a_{j+3k+1}\), \(i=1,\ldots,j\), so that the integrals 30 , become \[\label{Fh} \overline{F}_i=P_{2i-1,2i}^{a_{j+3k+1}}P_{2j+1,2(j+k)+1}^{-c_{i}}P_{2(j+k+1),n}^{c_i-a_{j+3k+1}}.\tag{41}\]

We note that the LV systems of type \([0,k,l]\) are a special case of the \([j,k,l]\) system.

3.2 Odd-dimensional inhomogeneous LV systems↩︎

For odd-dimensional \([j,k,l]\) LV systems, the vector \(\mathbf{r}=r\mathbf{1}\) is not in the image of \(\mathbf{A}\). We will consider vectors that are in the image of \(\mathbf{A}\) of the form \(\mathbf{r}=(r,\ldots,r,s)\) where \(s\) may depend on the parameters of the system. This means that, according to Theorem 1, homogeneous integrals of weight 0 that do not depend on \(x_n\) survive inhomogenisation.

3.2.1 The odd-dimensional inhomogeneous \([j,0,0]\) LV system↩︎

The \(n=2j+1\) dimensional inhomogeneous LV system of type \([j,0,0]\) has matrix \[\label{mnhj00oi} \mathbf{A}=\begin{pmatrix} J_1 & K_{1,2} & K_{1,3} & \cdots & K_{1,j} & L_1 \\ -K_{1,2} & J_2 & K_{2,3} & \cdots & K_{2,j} & L_2 \\ -K_{1,3} & - K_{2,3} & J_3 & \cdots & K_{3,j} & L_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ -K_{1,j} & -K_{2,j} & -K_{3,j} & \cdots & J_{j} & L_{j} \\ -L_1^T & -L_2^T & -L_3^T & \cdots & -L_{j}^T & J_{j+1} \end{pmatrix}\tag{42}\] where \[\label{JKLo} J_i=\begin{pmatrix} 0 & a_i \\ -a_i & 0 \end{pmatrix},\; K_{i,k}=(c_i-c_k)\mathbf{1}^{2,2},\; L_i=a_{j+1}\mathbf{1}^{2,1}.\tag{43}\] It is obtained by deleting the last row and column of ?? , and the substitution \(b_i=a_{j+1}\), \(i=1,\ldots,j\). We obtain 1 Casimir, \(\mathcal{C}=\prod_{i=1}^n x_i^{q_i}\), with \[q_i=\begin{cases} 1/a_m & i=2m-1,\quad m=1,\ldots,j+1, \\ -1/a_m & i=2m,\quad m=1,\ldots,j. \end{cases}\] The vector \(\mathbf{r}=r\mathbf{1}\) is not in the image of \(\mathbf{A}\), but we can take \[r_i=\begin{cases} r & i < n \\ 0 & i=n. \end{cases}\] The Hamiltonian is \(H=\sum_{i=1}^{2j+1} x_i + k_i \ln(x_i)\), with \[k_i=\begin{cases} -rq_i& i<n\\ 0 & i=n. \end{cases}\] and the functions \[\begin{align} R_i&=\mathcal{C}\left(\frac{F_i}{F_{i+1}}\right)^{1/(a_{j+1}(c_i-c_{i+1}))} \notag \\ &=\left(\frac{x_{2 i -1}+x_{2 i}}{x_{2i+1}+x_{2i+2}} \right)^{1/(c_i-c_{i+1})} \prod_{i=1}^j \left( \frac{x_{2 i -1}}{x_{2 i}}\right)^{1/a_i} \label{Ri} \end{align}\tag{44}\] are pairwise commuting integrals that do not depend on the variable \(x_n\). This yields the following result.

We note that there is one additional DP, \(P_{1,n-1}\), which yields an extra integral, \[\frac{P_{1,n-1}}{P_{1,2}}\prod_{i=2}^j \left( \frac{x_{2 i -1}}{x_{2 i}}\right)^{(c_i-c_1)/a_i}.\]

3.2.2 The odd-dimensional inhomogeneous \([0,k,0]\) LV system↩︎

To the homogeneous \(n=2k+1\) dimensional \([0,k,0]\) LV system we can add a inhomogeneous term with \[\label{bbb} r_i=\begin{cases} r & i<n \\ r\frac{a_{3k-1} - a_{3k}}{a_{3k-2}} & i=n. \end{cases}\tag{45}\] The Hamiltonian is \[H=\sum_{i=1}^{2j+1} x_i + r\frac{a_{3k} - a_{3k-1}}{a_{3k-2}a_{3k}} \ln(x_{n-1}) +\frac{r}{a_{3k}} \ln(x_{n}).\] Note that the highest index \(h\) of \(x_h\) in the function \(G_j\) as defined by ?? is \(2j+1<n\) for \(j\leq k-1\).

3.2.3 The odd-dimensional inhomogeneous \([0,k,l]\) LV system↩︎

To the \(n=2(k+l)+1\) dimensional \([0,k,l]\) LV system (\(l\neq0\)) we can add a inhomogeneous term with \[\label{nonht} r_i=\begin{cases} r & i<n \\ r\frac{a_{3k+1} - a_{3(k+l)-1}}{a_{3k+1}} & i=n. \end{cases}\tag{46}\]

3.2.4 The odd-dimensional inhomogeneous \([j,k,l]\) LV system↩︎

The matrix of the inhomogeneous \(n=2(j+k+l)+1\) dimensional \([j,k,l]\) LV system is obtained from the inhomogeneous even-dimensional \([j,k,l]\) system described in section 3.1.2, by deleting the last row and column. The matrix is obtained from the homogeneous case by substitution of \(b_i=c_i-a_{j+3k+1}\). The inhomogeneous term is given by \[\label{nonhtjkl} r_i=\begin{cases} r & i<n \\ r\frac{a_{j+3k+1} - a_{j+3(k+l)-1}}{a_{j+3k+1}} & i=n. \end{cases}\tag{47}\]

We note that the function \(K\), given by ?? , is not an integral for the inhomogeneous odd \([j,k,l]\) system. The number of integrals for the general odd \([j,k,l]\) system, with inhomogeneous term 47 , given in Prop. 1 is not sufficient for Liouville integrability. For the odd-dimensional \([j,k,0]\) system, with inhomogeneous term \[r_i=\begin{cases} r & i<n \\ r\frac{a_{j+3k-1} - a_{j+3k}}{a_{j+3k-2}} & i=n, \end{cases}\] we have also not been able to construct sufficiently many commuting integrals. There may be other inhomogeneous terms, i.e. not of the form \(\mathbf{r}=(r,\ldots,r,s)\) for which the number of integrals is sufficient. An example is given in Section 4.4, item 2.

4 Explicit examples of Liouville integrable LV systems↩︎

The number of triples \([j,k,l]\) with \(k\geq l\) and \(j+k+l=m-1\) is \[P(m)=\frac{m^{2}}{4}+\frac{m}{2}+\frac{1-\left(-1\right)^{m}}{8}.\] whereas the number of triples \([j,k,l]\) with \(k\geq l\), \(jk=0\) and \(j+k+l=m-1\) is \[Q(m)=\frac{m}{2}+\frac{7+(-1)^m}{4}.\] The values of \(P(m)\) and \(Q(m)\) for \(m\leq 10\) are given in Table [tabl].

We illustrate the theory by providing explicitly the families of Lotka-Volterra systems of type \([j,k,l]\), in dimensions \(n=3,4,5,6\). These correspond to \(n=2m\) or \(n=2m-1\) dimensional systems with \(j+k+l=m-1=1,2\), i.e. for \(m=2\): \[[j,k,l]=[1,0,0] \text{ or } [0,1,0],\] and for \(m=3\): \[[j,k,l]=[2,0,0], [1,1,0], [0,1,1] \text{ or } [0,2,0].\] In the following sections, each homogeneous system is understood to have Hamiltonian \(H=x_1+\cdots+x_n\). For the inhomogeneous systems the Hamiltonians will be given.

4.1 The even \([j,k,l]\) LV systems with \(j+k+l=1\) (\(n=4\))↩︎

1. The homogeneous [1,0,0] system has matrix \[\mathbf{A}=\left[\begin{array}{cccc} 0 & a_{1} & b_{1} & c_{1} \\ -a_{1} & 0 & b_{1} & c_{1} \\ -b_{1} & -b_{1} & 0 & a_{2} \\ -c_{1} & -c_{1} & -a_{2} & 0 \end{array}\right]\] and integral \[\left(x_{1}+x_{2}\right)^{a_{2}} x_{3}^{-c_{1}} x_{4}^{b_{1}}.\] One can add the vector \(\mathbf{r}=(r,r,r,r)\) if one takes \(b_1=c_1-a_2\).

2. The homogeneous [0,1,0] system has matrix \[\mathbf{A}=\left[\begin{array}{cccc} 0 & a_{1} & a_{2} & a_{4} \\ -a_{1} & 0 & a_{3} & a_{4} \\ -a_{2} & -a_{3} & 0 & a_{4} \\ -a_{4} & -a_{4} & -a_{4} & 0 \end{array}\right]\] and integral \[x_{1}^{-a_{3}} x_{2}^{a_{2}} x_{3}^{-a_{1}} \left(x_{1}+x_{2}+x_{3}\right)^{a_{1}-a_{2}+a_{3}}.\] The vector \(\mathbf{r}=(r,r,r,r)\) can be added without any further constraints.

4.2 The odd \([j,k,l]\) LV systems with \(j+k+l=1\) (\(n=3\))↩︎

1. The homogeneous [1,0,0] system has matrix \[\mathbf{A}=\left[\begin{array}{ccc} 0 & a_{1} & b_{1} \\ -a_{1} & 0 & b_{1} \\ -b_{1} & -b_{1} & 0 \end{array}\right]\] and Casimir \(x_{1}^{b_{1}} x_{2}^{-b_{1}} x_{3}^{a_{1}}\). We can add the inhomogeneous term \(\mathbf{r}=(r,r,0)\) and then the system has Hamiltonian \[H=x_{1}+x_{2}+x_{3}+\frac{r}{a_{1}} \ln\left(\frac{x_{2}}{x_{1}}\right).\]

2. The homogeneous [0,1,0] system has matrix \[\mathbf{A}=\left[\begin{array}{ccc} 0 & a_{1} & a_{2} \\ -a_{1} & 0 & a_{3} \\ -a_{2} & -a_{3} & 0 \end{array}\right],\] and contains the \([1,0,0]\) system as a special case. It has Casimir \(x_{1}^{a_{3}} x_{2}^{-a_{2}} x_{3}^{a_{1}}\), and inhomogeneous generalisation \(\mathbf{r}=\left(r,r,r\frac{a_{2}-a_{3}}{a_{1}}\right)\) with Hamiltonian \[H=x_{1}+x_{2}+x_{3}-r\frac{a_{2}-a_{3}}{a_{3} a_{1}} \ln \left(x_{2}\right)+\frac{r}{a_{3}} \ln\left(x_{3}\right).\]

4.3 The even \([j,k,l]\) LV systems with \(j+k+l=2\) (\(n=6\))↩︎

1. The homogeneous [2,0,0] LV system has matrix \[\mathbf{A}=\left[\begin{array}{cccccc} 0 & a_{1} & \frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & \frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & b_{1} & c_{1} \\ -a_{1} & 0 & \frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & \frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & b_{1} & c_{1} \\ -\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & -\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & 0 & a_{2} & b_{2} & c_{2} \\ -\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & -\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & -a_{2} & 0 & b_{2} & c_{2} \\ -b_{1} & -b_{1} & -b_{2} & -b_{2} & 0 & a_{3} \\ -c_{1} & -c_{1} & -c_{2} & -c_{2} & -a_{3} & 0 \end{array}\right]\] and commuting integrals \[\left(x_{1}+x_{2}\right)^{a_{3}} x_{5}^{-c_{1}} x_{6}^{b_{1}},\quad \left(x_{3}+x_{4}\right)^{a_{3}} x_{5}^{-c_{2}} x_{6}^{b_{2}}.\] To the subsystem with \(b_1=c_1-a_3,b_2=c_2-a_3\), which has matrix \[\left[\begin{array}{cccccc} 0 & a_{1} & c_{1}-c_{2} & c_{1}-c_{2} & c_{1}-a_{3} & c_{1} \\ -a_{1} & 0 & c_{1}-c_{2} & c_{1}-c_{2} & c_{1}-a_{3} & c_{1} \\ -c_{1}+c_{2} & -c_{1}+c_{2} & 0 & a_{2} & c_{2}-a_{3} & c_{2} \\ -c_{1}+c_{2} & -c_{1}+c_{2} & -a_{2} & 0 & c_{2}-a_{3} & c_{2} \\ -c_{1}+a_{3} & -c_{1}+a_{3} & -c_{2}+a_{3} & -c_{2}+a_{3} & 0 & a_{3} \\ -c_{1} & -c_{1} & -c_{2} & -c_{2} & -a_{3} & 0 \end{array}\right]\] one can add a inhomogeneous term \(\mathbf{r}=(r,r,r,r,r,r)\). It has Hamiltonian \[H=x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+\frac{r}{a_{3}} \ln\left(\frac{x_{6}}{x_{5}}\right),\] and commuting integrals \(\left(x_{1}+x_{2}\right)^{a_{3}} x_{5}^{-c_{1}} x_{6}^{c_{1}-a_{3}}\) and \(\left(x_{3}+x_{4}\right)^{a_{3}} x_{5}^{-c_{2}} x_{6}^{c_{2}-a_{3}}\).

2. The homogeneous [1,1,0] system has matrix \[\mathbf{A}=\left[\begin{array}{cccccc} 0 & a_{1} & b_{1} & b_{1} & b_{1} & c_{1} \\ -a_{1} & 0 & b_{1} & b_{1} & b_{1} & c_{1} \\ -b_{1} & -b_{1} & 0 & a_{2} & a_{3} & a_{5} \\ -b_{1} & -b_{1} & -a_{2} & 0 & a_{4} & a_{5} \\ -b_{1} & -b_{1} & -a_{3} & -a_{4} & 0 & a_{5} \\ -c_{1} & -c_{1} & -a_{5} & -a_{5} & -a_{5} & 0 \end{array}\right]\] and commuting integrals \[\left(x_{1}+x_{2}\right)^{a_{5}} \left(x_{3}+x_{4}+x_{5}\right)^{-c_{1}} x_{6}^{b_{1}},\quad x_{3}^{-a_{4}} x_{4}^{a_{3}} x_{5}^{-a_{2}} \left(x_{3}+x_{4}+x_{5}\right)^{a_{2}-a_{3}+a_{4}}.\] The inhomogeneous [1,1,0] system has matrix \[\mathbf{A}=\left[\begin{array}{cccccc} 0 & a_{1} & c_{1}-a_{5} & c_{1}-a_{5} & c_{1}-a_{5} & c_{1} \\ -a_{1} & 0 & c_{1}-a_{5} & c_{1}-a_{5} & c_{1}-a_{5} & c_{1} \\ -c_{1}+a_{5} & -c_{1}+a_{5} & 0 & a_{2} & a_{3} & a_{5} \\ -c_{1}+a_{5} & -c_{1}+a_{5} & -a_{2} & 0 & a_{4} & a_{5} \\ -c_{1}+a_{5} & -c_{1}+a_{5} & -a_{3} & -a_{4} & 0 & a_{5} \\ -c_{1} & -c_{1} & -a_{5} & -a_{5} & -a_{5} & 0 \end{array}\right],\quad \mathbf{r}=\left[\begin{array}{c} r\\ r\\ r\\ r\\ r\\ r \end{array}\right],\] Hamiltonian \[x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+\frac{b}{a_5} \left(\frac{-a_{4} \ln\left(x_{3}\right)+a_{3} \ln\left(x_{4}\right)-a_{2} \ln\left(x_{5}\right)}{ a_{2}-a_{3}+a_{4}}+\ln\left(x_{6}\right)\right)\] and integrals \(\left(x_{1}+x_{2}\right)^{a_{5}} \left(x_{3}+x_{4}+x_{5}\right)^{-c_{1}} x_{6}^{c_{1}-a_{5}}\), \(x_{3}^{-a_{4}} x_{4}^{a_{3}} x_{5}^{-a_{2}} \left(x_{3}+x_{4}+x_{5}\right)^{a_{2}-a_{3}+a_{4}}\).

3. The homogeneous [0,1,1] system has matrix \[\mathbf{A}=\left[\begin{array}{cccccc} 0 & a_{1} & a_{2} & a_{4} & a_{4} & a_{4} \\ -a_{1} & 0 & a_{3} & a_{4} & a_{4} & a_{4} \\ -a_{2} & -a_{3} & 0 & a_{4} & a_{4} & a_{4} \\ -a_{4} & -a_{4} & -a_{4} & 0 & a_{5} & a_{6} \\ -a_{4} & -a_{4} & -a_{4} & -a_{5} & 0 & a_{7} \\ -a_{4} & -a_{4} & -a_{4} & -a_{6} & -a_{7} & 0 \end{array}\right]\] and commuting integrals \[x_{1}^{-a_{3}} x_{2}^{a_{2}} x_{3}^{-a_{1}} \left(x_{1}+x_{2}+x_{3}\right)^{a_{1}-a_{2}+a_{3}},\quad x_{4}^{-a_{7}} x_{5}^{a_{6}} x_{6}^{-a_{5}} \left(x_{4}+x_{5}+x_{6}\right)^{a_{5}-a_{6}+a_{7}}.\] We can allow the inhomogeneous term with \(\mathbf{r}=(r,r,r,r,r,r)\), in which case the Hamiltonian is \[\begin{align} H&=x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+\frac{r}{a_4} \left(\frac{-a_{3} \ln\left(x_{1}\right)+a_{2} \ln\left(x_{2}\right)-a_{1} \ln\left(x_{3}\right)}{ a_{1}-a_{2}+a_{3}}\right.\\ &\;\;\;\;\;\;\;\;\;\left.+\frac{a_{7} \ln\left(x_{4}\right)-a_{6} \ln\left(x_{5}\right)+a_{5} \ln\left(x_{6}\right)}{ a_{5}-a_{6}+a_{7}} \right). \end{align}\]

4. The homogeneous [0,2,0] system has matrix \[\mathbf{A}=\left[\begin{array}{cccccc} 0 & a_{1} & a_{2} & a_{4} & a_{5} & a_{7} \\ -a_{1} & 0 & a_{3} & a_{4} & a_{5} & a_{7} \\ -a_{2} & -a_{3} & 0 & a_{4} & a_{5} & a_{7} \\ -a_{4} & -a_{4} & -a_{4} & 0 & a_{6} & a_{7} \\ -a_{5} & -a_{5} & -a_{5} & -a_{6} & 0 & a_{7} \\ -a_{7} & -a_{7} & -a_{7} & -a_{7} & -a_{7} & 0 \end{array}\right]\] and commuting integrals \[\frac{x_{2}^{a_{2}} \left(x_{1}+x_{2}+x_{3}\right)^{a_{1}-a_{2}+a_{3}}}{x_{1}^{a_{3}}x_{3}^{a_{1}} },\quad \frac{ x_{4}^{a_{5}} \left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}\right)^{a_{4}-a_{5}+a_{6}} }{\left(x_{1}+x_{2}+x_{3}\right)^{a_{6}}x_{5}^{a_{4}} }.\]

We can allow the inhomogeneous term with \(\mathbf{r}=(r,r,r,r,r,r)\), in which case the Hamiltonian is \[\begin{align} H&=x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+\frac{r}{a_7} \left(\frac{a_6(-a_{3} \ln\left(x_{1}\right)+a_{2} \ln\left(x_{2}\right)-a_{1} \ln\left(x_{3}\right))}{ \left(a_{1}-a_{2}+a_{3}\right) \left(a_{4}-a_{5}+a_{6}\right)}\right.\\ &\;\;\;\;\;\;\;\;\;\left.+\frac{a_{5} \ln\left(x_{4}\right)-a_{4} \ln\left(x_{5}\right)}{ a_{4}-a_{5}+a_{6}} + \ln\left(x_{6}\right) \right). \end{align}\]

4.4 The odd \([j,k,l]\) LV systems with \(j+k+l=2\) (\(n=5\))↩︎

1. The homogeneous [2,0,0] system has matrix \[\left[\begin{array}{ccccc} 0 & a_{1} & \frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & \frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & b_{1} \\ -a_{1} & 0 & \frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & \frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & b_{1} \\ -\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & -\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & 0 & a_{2} & b_{2} \\ -\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & -\frac{b_{1} c_{2}-b_{2} c_{1}}{a_{3}} & -a_{2} & 0 & b_{2} \\ -b_{1} & -b_{1} & -b_{2} & -b_{2} & 0 \end{array}\right],\] integral \(\dfrac{\left(x_{1}+x_{2}\right)^{b_{2}}}{ \left(x_{3}+x_{4}\right)^{b_{1}}}x_{5}^{\frac{c_{2} b_{1}-c_{1} b_{2}}{a_{3}}}\) and Casimir \(\left(\dfrac{x_{1}}{x_{2}}\right)^{b_{1} a_{2}} \left(\dfrac{x_{3}}{x_{4}}\right)^{a_{1} b_{2}} x_{5}^{a_{1} a_{2}}\).

   The inhomogeneous [2,0,0] system, with \(\mathbf{r}=(r,r,r,r,0)\), has matrix \[\mathbf{A}=\left[\begin{array}{ccccc} 0 & a_{1} & c_{2}-c_{1} & c_{2}-c_{1} & a_{3} \\ -a_{1} & 0 & c_{2}-c_{1} & c_{2}-c_{1} & a_{3} \\ c_{1}-c_{2} & c_{1}-c_{2} & 0 & a_{2} & a_{3} \\ c_{1}-c_{2} & c_{1}-c_{2} & -a_{2} & 0 & a_{3} \\ -a_{3} & -a_{3} & -a_{3} & -a_{3} & 0 \end{array}\right],\] Hamiltonian \[H=x_{1}+x_{2}+x_{3}+x_4+x_5+\frac{r}{a_{1}} \ln\left(\frac{x_{2}}{x_{1}}\right)+\frac{r}{a_{2}} \ln\left(\frac{x_{4}}{x_{3}}\right),\] integrals \[\left(\frac{x_{1}+x_{2}}{x_{3}+x_{4}}\right)^{1/(c_1-c_2)} \left(\frac{x_{1}}{x_{2}}\right)^{1/a_1} \left(\frac{x_{3}}{x_{4}}\right)^{1/a_2} ,\quad \frac{x_{1}+x_{2}+x_{3}+x_{4}}{x_{1}+x_{2}} \left(\frac{x_{3}}{x_{4}}\right)^{(c_2-a_1)/a_2}\] and Casimir \(\left(\dfrac{x_{1}}{x_{2}}\right)^{1/a_{1}} \left(\dfrac{x_{3}}{x_{4}}\right)^{1/a_2} x_{5}^{1/a_3}\).

2. The homogeneous [1,1,0] system has matrix \[\mathbf{A}=\left[\begin{array}{ccccc} 0 & a_{1} & b_{1} & b_{1} & b_{1} \\ -a_{1} & 0 & b_{1} & b_{1} & b_{1} \\ -b_{1} & -b_{1} & 0 & a_{2} & a_{3} \\ -b_{1} & -b_{1} & -a_{2} & 0 & a_{4} \\ -b_{1} & -b_{1} & -a_{3} & -a_{4} & 0 \end{array}\right],\] integral \(\frac{x_{4}^{a_{3}}\left(x_{3}+x_{4}+x_{5}\right)^{a_{2}-a_{3}+a_{4}}}{x_{3}^{a_{4}}x_{5}^{a_{2}} }\) and Casimir \(\left(\frac{x_{1}}{x_2}\right)^{\frac{b_{1}}{a_{1}}} \left(\frac{x_{3}^{a_{4}}x_{5}^{a_{2}}}{x_{4}^{a_{3}}}\right)^{1/(a_{2}-a_{3}+a_{4})}\). One can not add a inhomogeneous term of the form \(\mathbf{r}=(r,\ldots,r,s)\). However, one can take \(\mathbf{r}=(r,r,0,0,0)\) (without constraints on the parameters) and then the Hamiltonian is \[H=x_1+x_2+x_3+x_4+x_5+\frac{r}{a_1}\ln\left(\frac{x_2}{x_1}\right).\]

3. The [0,1,1] system has matrix \[\mathbf{A}=\left[\begin{array}{ccccc} 0 & a_{1} & a_{2} & a_{4} & a_{4} \\ -a_{1} & 0 & a_{3} & a_{4} & a_{4} \\ -a_{2} & -a_{3} & 0 & a_{4} & a_{4} \\ -a_{4} & -a_{4} & -a_{4} & 0 & a_{5} \\ -a_{4} & -a_{4} & -a_{4} & -a_{5} & 0 \end{array}\right],\] integral \(\frac{x_{2}^{a_{2}}\left(x_{1}+x_{2}+x_{3}\right)^{a_{1}-a_{2}+a_{3}} }{x_{1}^{a_{3}}x_{3}^{a_{1}}}\) and Casimir \(\left(\frac{x_{1}^{a_{3}}x_{3}^{a_{1}}}{x_{2}^{a_{2}}}\right)^{a_{5}} \left(\frac{x_{5}}{x_{4}}\right)^{a_{4}(a_{1}-a_{2}+a_{3})}\). The Hamiltonian \(H=x_1+x_2+x_3+x_4+x_5+r\frac{a_5-a_4}{a_4a_5}\ln(x_4)+\frac{r}{a_5}\ln(x_5)\) yields a inhomogeneous term with \(\mathbf{r}=\left(r,r,r,r,r\frac{a_4-a_5}{a_4}\right)\).

4. The [0,2,0] system (which contains the [0,1,1] case) has matrix \[\mathbf{A}=\left[\begin{array}{ccccc} 0 & a_{1} & a_{2} & a_{4} & a_{5} \\ -a_{1} & 0 & a_{3} & a_{4} & a_{5} \\ -a_{2} & -a_{3} & 0 & a_{4} & a_{5} \\ -a_{4} & -a_{4} & -a_{4} & 0 & a_{6} \\ -a_{5} & -a_{5} & -a_{5} & -a_{6} & 0 \end{array}\right],\] integral \(\frac{x_{2}^{a_{2}}\left(x_{1}+x_{2}+x_{3}\right)^{a_{1}-a_{2}+a_{3}} }{x_{1}^{a_{3}}x_{3}^{a_{1}}}\) and Casimir \(\left(\frac{x_{1}^{a_{3}}x_{3}^{a_{1}}}{x_{2}^{a_{2}}}\right)^{a_6} \left(\frac{x_{5}^{a_{4}}}{x_{4}^{a_{5}}}\right)^{a_{1}-a_{2}+a_{3}}\). The inhomogeneous term is \(\mathbf{r}=\left(r,r,r,r, r\frac{\left(a_{5}-a_{6}\right)}{a_{4}}\right)\) with Hamiltonian \(H=x_1+x_2+x_3+x_4+x_5+b \frac{a_{6}-a_{5}}{a_{4} a_{6}} \ln\left(x_{4}\right)+\frac{b}{a_{6}}\ln\left(x_{5}\right)\).

4.5 Examples of Liouville integrable LV systems not of type \([j,k,l]\)↩︎

There exist Liouville integrable LV systems with associated hypergraphs which are not in the \([j,k,l]\) family displayed in Figure [F0]. These systems seem to have less parameters, and additional DPs and/or integrals.

In this section, the hyperedges correspond to the DPs of the system, not to associated commuting integrals (which does not even make sense when the matrix \(\mathbf{A}\) is singular). We give three examples of \(10\)-component classes of LV systems.

1. The 10-component LV system with rank 8 matrix \[\label{SM} \mathbf{A}=\begin{pmatrix} 0 & a_{1} & a_{1} & a_{1} & a_{2} & a_{2} & a_{2} & a_{3} & a_{3} & a_{3} \\ -a_{1} & 0 & a_{1} & a_{1} & a_{2} & a_{2} & a_{2} & a_{3} & a_{3} & a_{3} \\ -a_{1} & -a_{1} & 0 & 0 & a_{2} & a_{2} & a_{2} & a_{3} & a_{3} & a_{3} \\ -a_{1} & -a_{1} & 0 & 0 & a_{2} & a_{2} & a_{2} & a_{3} & a_{3} & a_{3} \\ -a_{2} & -a_{2} & -a_{2} & -a_{2} & 0 & a_{4} & a_{5} & a_{7} & a_{7} & a_{7} \\ -a_{2} & -a_{2} & -a_{2} & -a_{2} & -a_{4} & 0 & a_{6} & a_{7} & a_{7} & a_{7} \\ -a_{2} & -a_{2} & -a_{2} & -a_{2} & -a_{5} & -a_{6} & 0 & a_{7} & a_{7} & a_{7} \\ -a_{3} & -a_{3} & -a_{3} & -a_{3} & -a_{7} & -a_{7} & -a_{7} & 0 & a_{8} & a_{9} \\ -a_{3} & -a_{3} & -a_{3} & -a_{3} & -a_{7} & -a_{7} & -a_{7} & -a_{8} & 0 & a_{10} \\ -a_{3} & -a_{3} & -a_{3} & -a_{3} & -a_{7} & -a_{7} & -a_{7} & -a_{9} & -a_{10} & 0 \end{pmatrix}\tag{48}\] admits (apart from the Hamiltonian \(H=\sum_{i=1}^{10} x_i\)) the following DPs \[x_{1}+x_{2}, x_{3}+x_{4}, x_{2}+x_{3}+x_{4}, x_{5}+x_{6}+x_{7}, x_{8}+x_{9}+x_{10} , x_{1}+x_{2}+x_{3}+x_{4},\] that is, it has the associated hypergraph depicted in Fig. [F5].

   

   The matrix 48 has rank \(10-2=8\). The functions \[\frac{x_{3}}{x_{4}}, \quad x_{1}^{-a_{7}} x_{2}^{a_{7}} x_{3}^{-a_{7}} x_{5}^{\frac{a_{3} a_{6}}{a_{4}-a_{5}+a_{6}}} x_{6}^{-\frac{a_{3} a_{5}}{a_{4}-a_{5}+a_{6}}} x_{7}^{\frac{a_{3} a_{4}}{a_{4}-a_{5}+a_{6}}} x_{8}^{-\frac{a_{10} a_{2}}{a_{8}-a_{9}+a_{10}}} x_{9}^{\frac{a_{2} a_{9}}{a_{8}-a_{9}+a_{10}}} x_{10}^{-\frac{a_{8} a_{2}}{a_{8}-a_{9}+a_{10}}}\] are Casimirs, and the set of integrals \(H\), \[\frac{\left(x_{1}+x_{2}+x_{3}+x_{4}\right) x_{2}}{x_{1} x_{3}},\quad \frac{\left(x_{5}+x_{6}+x_{7}\right)^{a_{4}-a_{5}+a_{6}} x_{6}^{a_{5}}}{x_{5}^{a_{6}} x_{7}^{a_{4}}},\quad \frac{\left(x_{8}+x_{9}+x_{10}\right)^{a_{8}-a_{9}+a_{10}} x_{9}^{a_{9}} }{x_{8}^{a_{10}} x_{10}^{a_{8}}}\] is pairwise commuting. Although the hypergraph in Fig. [F5] contains the type \([2,1,1]\) hypergraph, the LV system with matrix 48 is not a subsystem of the corresponding LV system.

2. The LV system with rank 8 matrix \[\label{ansa} \mathbf{A}=\begin{pmatrix} 0 & a_{1} & a_{2} & a_{2} & a_{3} & a_{3} & a_{3} & a_{3} & a_{3} & a_{3} \\ -a_{1} & 0 & a_{2} & a_{2} & a_{3} & a_{3} & a_{3} & a_{3} & a_{3} & a_{3} \\ -a_{2} & -a_{2} & 0 & a_{4} & a_{5} & a_{5} & a_{5} & a_{5} & a_{5} & a_{5} \\ -a_{2} & -a_{2} & -a_{4} & 0 & a_{5} & a_{5} & a_{5} & a_{5} & a_{5} & a_{5} \\ -a_{3} & -a_{3} & -a_{5} & -a_{5} & 0 & a_{6} & a_{7} & a_{7} & a_{7} & a_{7} \\ -a_{3} & -a_{3} & -a_{5} & -a_{5} & -a_{6} & 0 & a_{7} & a_{7} & a_{7} & a_{7} \\ -a_{3} & -a_{3} & -a_{5} & -a_{5} & -a_{7} & -a_{7} & 0 & a_{7} & a_{7} & a_{7} \\ -a_{3} & -a_{3} & -a_{5} & -a_{5} & -a_{7} & -a_{7} & -a_{7} & 0 & a_{8}-a_{9} & a_{8} \\ -a_{3} & -a_{3} & -a_{5} & -a_{5} & -a_{7} & -a_{7} & -a_{7} & -a_{8}+a_{9} & 0 & a_{9} \\ -a_{3} & -a_{3} & -a_{5} & -a_{5} & -a_{7} & -a_{7} & -a_{7} & -a_{8} & -a_{9} & 0 \end{pmatrix}\tag{49}\] admits the DPs \[x_{1}+x_{2},\;x_{3}+x_{4},\;x_{5}+x_{6},\;x_{5}+x_{6}+x_{7},\; x_{8}+x_{9}+x_{10},\;x_{7}+x_{8}+x_{9}+x_{10},\; x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10},\] see associated hypergraph in Fig. [ahsa].

   

   It possesses 8 functionally independent integrals \[\begin{align} H &= x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}\\ C_{1} &= \left(\frac{x_{1}}{x_{2}}\right)^{\frac{a_{3}}{a_{1}}} \left(\frac{x_{3}}{x_{4}}\right)^{\frac{a_{5}}{a_{4}}} \left(\frac{x_{5}}{x_{6}}\right)^{\frac{a_{7}}{a_{6}}} x_{7}\left(\frac{x_{10}}{x_{8}}\right)^{\frac{a_{7}}{a_{8}}},\quad C_{2} = \frac{x_{9}^{a_{8}} x_{10}^{a_{9}-a_{8}}}{x_{8}^{a_{9}}} \\ R_{1} &=\left(\frac{x_{2}}{x_{1}}\right)^{\frac{a_{3}}{a_{1}}} \left(\frac{x_{4}}{x_{3}}\right)^{\frac{a_{5}}{a_{4}}} \frac{\left(x_{1}+x_{2}\right)^{\frac{a_{5}}{a_{2}}}}{ \left(x_{3}+x_{4}\right)^{\frac{a_{3}}{a_{2}}}} \\ R_{2} &= \left(\frac{x_{1}}{x_{2}}\right)^{\frac{a_{3}}{a_{1}}} \left(\frac{x_{3}}{x_{4}}\right)^{\frac{a_{5}}{a_{4}}} \left(x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}\right) \\ R_{3} &= \frac{\left(x_{5}+x_{6}+x_{7}\right)^{a_{6}} x_{6}^{a_{7}}}{x_{5}^{a_{7}}x_{7}^{a_{6}}},\quad R_{4} = \frac{\left(x_{7}+x_{8}+x_{9}+x_{10}\right)^{a_{8}} x_{8}^{a_{7}}}{x_{7}^{a_{8}}x_{10}^{a_{7}}} \\ R_{5} &= \left(\frac{x_{1}}{x_{2}}\right)^{\frac{a_{2}}{a_{1}}} \left(x_{3}+x_{4}\right) \left(\frac{x_{6}}{x_{5}}\right)^{\frac{a_{5}}{a_{6}}} \left(\frac{x_{8}+x_{9}+x_{10}}{x_{7}}\right)^{\frac{a_{5}}{a_{7}}}, \end{align}\] where \(C_1,C_2\) are Casimirs, and the following brackets vanish \[\{R_1,R_2\}=\{R_1,R_3\}=\{R_1,R_4\}=\{R_2,R_3\}=\{R_2,R_4\}=\{R_3,R_5\}=0.\]

3. The LV system with nonsingular matrix \[\label{lastone} \mathbf{A}=\begin{pmatrix} 0 & a_{1} & a_{2} & a_{4} & a_{4} & a_{4} & a_{5} & a_{5} & a_{5} & a_{5} \\ -a_{1} & 0 & a_{3} & a_{4} & a_{4} & a_{4} & a_{5} & a_{5} & a_{5} & a_{5} \\ -a_{2} & -a_{3} & 0 & a_{4} & a_{4} & a_{4} & a_{5} & a_{5} & a_{5} & a_{5} \\ -a_{4} & -a_{4} & -a_{4} & 0 & a_{6} & a_{7} & a_{9} & a_{9} & a_{9} & a_{9} \\ -a_{4} & -a_{4} & -a_{4} & -a_{6} & 0 & a_{8} & a_{9} & a_{9} & a_{9} & a_{9} \\ -a_{4} & -a_{4} & -a_{4} & -a_{7} & -a_{8} & 0 & a_{9} & a_{9} & a_{9} & a_{9} \\ -a_{5} & -a_{5} & -a_{5} & -a_{9} & -a_{9} & -a_{9} & 0 & a_{11} & a_{11} & a_{11} \\ -a_{5} & -a_{5} & -a_{5} & -a_{9} & -a_{9} & -a_{9} & -a_{11} & 0 & a_{10} & a_{11} \\ -a_{5} & -a_{5} & -a_{5} & -a_{9} & -a_{9} & -a_{9} & -a_{11} & -a_{10} & 0 & a_{11} \\ -a_{5} & -a_{5} & -a_{5} & -a_{9} & -a_{9} & -a_{9} & -a_{11} & -a_{11} & -a_{11} & 0 \end{pmatrix}\tag{50}\] admits the DPs \[x_{1}+x_{2}+x_{3}, x_{4}+x_{5}+x_{6}, x_{7}+x_{8}+x_{9} , x_{8}+x_{9}, x_{8}+x_{9}+x_{10}, x_{7}+x_{8}+x_{9}+x_{10},\] as depicted in Fig. [F7].

   

   The system has the following set of pairwise commuting integrals \[\begin{align} T_1&=\left(x_{1}+x_{2}+x_{3}\right)^{-a_{1}+a_{2}-a_{3}} x_{1}^{a_{3}} x_{2}^{-a_{2}} x_{3}^{a_{1}} ,\quad T_2=\left(x_{4}+x_{5}+x_{6}\right)^{-a_{6}+a_{7}-a_{8}} x_{4}^{a_{8}} x_{5}^{-a_{7}} x_{6}^{a_{6}},\\ T_3&= \frac{\left(x_{7}+x_{8}+x_{9}\right) \left(x_{8}+x_{9}+x_{10}\right)}{x_{7} x_{10}},\quad H=x_{1}+x_{2}+\cdots+x_{10},\\ T_4&= \left(x_{8}+x_{9}\right)^{-a_{4}} x_{1}^{\frac{a_{9} a_{3}}{a_{1}-a_{2}+a_{3}}} x_{2}^{-\frac{a_{2} a_{9}}{a_{1}-a_{2}+a_{3}}} x_{3}^{\frac{a_{1} a_{9}}{a_{1}-a_{2}+a_{3}}} x_{4}^{-\frac{a_{8} a_{5}}{a_{6}-a_{7}+a_{8}}} x_{5}^{\frac{a_{5} a_{7}}{a_{6}-a_{7}+a_{8}}} x_{6}^{-\frac{a_{6} a_{5}}{a_{6}-a_{7}+a_{8}}} x_{7}^{a_{4}} x_{10}^{a_{4}}, \end{align}\] and 1 additional integral \[T_5=x_{8}^{-a_{11}} x_{9}^{a_{11}} x_{10}^{-a_{10}} \left(x_{8}+x_{9}+x_{10}\right)^{a_{10}}\] which commutes with \(T_1\) and \(T_2\).

   The system is not equivalent to a subsystem of the 10-component LV system of type \([2,1,1]\).

We note that the linear transformations that preserve the Lotka-Volterra structure (and induce the equivalence relation introduced in [23]) do not preserve the antisymmetry of the matrix \(\mathbf{A}\).

Acknowledgement. We are grateful for the hospitality of Da-jun Zhang and Cheng Zhang of Shanghai University, where this paper was finalised. P.H. van der Kamp was supported by NSFC grant No. 12271334. We thank Ian Marquette for helpful discussions.

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