On generalized Volterra systems

TL;DR

This paper introduces a broad class of integrable Hamiltonian systems extending the KM system, including transformations to Lotka-Volterra systems, with detailed examples in low dimensions and generalizations to complex Lie algebras.

Contribution

It constructs new integrable Hamiltonian systems generalizing the KM system and provides explicit examples and transformations in dimensions 4 and 5, extending to complex Lie algebras.

Findings

Construction of a large family of integrable systems

Explicit examples in dimensions 4 and 5

Transformations to quadratic Lotka-Volterra systems

Abstract

We construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka-Volterra system. We present in detail all such systems in dimensions 4 and 5 and we also give some examples from higher dimensions. This construction generalizes easily to each complex simple Lie algebra.