The analytic integrability problem for perturbations of homogeneous quadratic Lotka-Volterra systems

TL;DR

This paper addresses the problem of determining when certain planar differential systems with quadratic Lotka-Volterra structure are analytically integrable, providing a complete solution for systems with specific homogeneous polynomial components.

Contribution

It offers a complete solution to the analytic integrability problem for quadratic Lotka-Volterra systems with homogeneous components, extending understanding of integrability conditions.

Findings

Characterization of integrable systems within the class studied

Explicit conditions for the existence of first integrals

Application to systems with polynomial perturbations

Abstract

We solve the analytic integrability problem for diferential systems in the plane whose origin is an isolated singularity and the first homogeneous component is a quadratic Lotka-Volterra type. As an application, we give the analytically integrable systems of a class of systems $\dot{x}= x(P1+P2)$; $\dot{y}= y(Q1+Q2)$; being Pi,Qi homogeneous polynomials of degree i.