Integrability and Linearizability of a Family of Three-Dimensional Polynomial Systems

TL;DR

This paper studies the conditions under which certain three-dimensional polynomial systems are integrable or linearizable, providing criteria, an algorithm, and applying it to a quadratic subfamily to enhance understanding of their integrability properties.

Contribution

It introduces a convergence criterion for the normal form, an efficient algorithm for integrability conditions, and applies these to a quadratic subfamily of systems.

Findings

Established a criterion for the convergence of the normal form.

Developed an algorithm to determine integrability conditions.

Analyzed a quadratic subfamily for integrability and linearizability.

Abstract

We investigate the local integrability and linearizability of a family of three-dimensional polynomial systems with the matrix of the linear approximation having the eigenvalues $1, \zeta, \zeta^2 $, where $\zeta$ is a primitive cubic root of unity. We establish a criterion for the convergence of the Poincar\'e--Dulac normal form of the systems and examine the relationship between the normal form and integrability. Additionally, we introduce an efficient algorithm to determine the necessary conditions for the integrability of the systems. This algorithm is then applied to a quadratic subfamily of the systems to analyze its integrability and linearizability. Our findings offer insights into the integrability properties of three-dimensional polynomial systems.