In a recent paper [1], completely integrable cases were discovered of the Lotka Volterra Hamiltonian (LVH) system without linear terms, $\frac{d{x}_i}{dt}={\dot{x}}_i={\sum }_{j=1}^n{a}_{ij}{x}_i{x}_j,i=1,\dots ,n$, $a_{ij}=-a_{ji}$, satisfying the condition $H={\sum }_{i=1}^n{x}_i=h=const$. In this paper, we first generalize this system to one that includes an arbitrary set of linear terms that preserve the Hamiltonian integral. We thus discover a wide class of LVH systems which we claim to be integrable, since their equations possess the Painlevé property, i.e. their solutions have only poles as movable singularities in the complex t-plane. Next, we focus on the case $n=3$ and vary some of the parameters, including additional nonlinearities to look for nonintegrable extensions with interesting dynamical properties. Our results suggest that, in this class of systems, non - integrability generally yields simple dynamics far removed from the type of complexity one expects from non - integrable 3 - dimensional nonlinear systems.

在最近的一篇论文中[1]，发现了Lotka Volterra Hamiltonian（LVH）系统的完全可积案例，其中没有线性项， $\frac{d{X}_{\mathrm{一世}}}{dŤ}={\dot{X}}_{\mathrm{一世}}={\sum }_{Ĵ=\mathrm{1个}}^{ñ}{\mathrm{一种}}_{\mathrm{一世}Ĵ}{X}_{\mathrm{一世}}{X}_{Ĵ}，\mathrm{一世}=\mathrm{1个}，\dots ，ñ$， ${\mathrm{一种}}_{\mathrm{一世}Ĵ}=-{\mathrm{一种}}_{Ĵ\mathrm{一世}}$，满足条件 $H={\sum }_{\mathrm{一世}=\mathrm{1个}}^{ñ}{X}_{\mathrm{一世}}=H=CØñsŤ$。在本文中，我们首先将此系统推广为包括保留汉密尔顿积分的任意线性项的系统。因此，我们发现了一大类LVH系统，我们声称它们是可积分的，因为它们的方程式具有Painlevé性质，即，它们的解仅具有极点，作为复t平面中的可移动奇点。接下来，我们关注案例$ñ=3$并更改一些参数，包括其他非线性，以寻找具有有趣动态特性的不可积分扩展。我们的结果表明，在这类系统中，不可积分性通常会产生简单的动力学，而与那些不可积分的三维非线性系统所期望的复杂性相去甚远。