Algebraic Integrability of Lotka-Volterra equations in three dimensions

TL;DR

This paper classifies three-dimensional Lotka-Volterra systems with skew symmetric matrices, establishing necessary and sufficient conditions for their algebraic complete integrability using Painleve analysis and Kowalevski exponents.

Contribution

It provides a complete classification of algebraically integrable 3D Lotka-Volterra systems based on Kowalevski exponents and integrality conditions.

Findings

Complete classification of integrable systems

Necessary and sufficient conditions identified

Use of Painleve analysis and Kowalevski exponents

Abstract

We examine the algebraic complete integrability of Lotka-Volterra equations in three dimensions. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painleve analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions for the algebraic integrability of the corresponding systems. We also show that the conditions are sufficient.