Planar polynomial vector fields having a polynomial first integral can be obtained from linear systems

TL;DR

This paper demonstrates that planar polynomial differential systems with a polynomial first integral can be derived from linear systems via polynomial transformations, providing a method to generate such systems.

Contribution

It establishes a constructive link between linear systems and polynomial vector fields with polynomial first integrals using polynomial changes of variables.

Findings

Polynomial systems with polynomial first integrals can be obtained from linear systems.

The method provides a systematic way to generate such systems.

This approach simplifies the analysis of polynomial vector fields with integrals.

Abstract

We consider in this work planar polynomial differential systems having a polynomial first integral. We prove that these systems can be obtained from a linear system through a polynomial change of variables.