On the Integrability Problem for the Hopf-Zero singularity and its relation with the inverse Jacobi multiplier

TL;DR

This paper investigates the conditions under which the Hopf-Zero singularity admits first integrals, exploring the connection with inverse Jacobi multipliers and providing algorithms for their detection.

Contribution

It introduces necessary conditions for first integrals in Hopf-Zero singularities and links their existence to inverse Jacobi multipliers, with practical algorithms for analysis.

Findings

Derived necessary conditions for first integrals

Established relation between first integrals and inverse Jacobi multipliers

Provided algorithms for detecting first integrals in specific vector fields

Abstract

In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first integrals are presented, and they are applied to some families of vector fields.