A Note on the First Integrals of Vector Fields with Integrating Factors and Normalizers

TL;DR

This paper establishes a geometric, coordinate-free criterion for the existence of explicit first integrals in vector fields with integrating factors, extending prior results on volume-preserving fields with normalizers across smooth manifolds.

Contribution

It provides a new sufficient condition for integrability of vector fields with integrating factors, generalizing previous results to any smooth orientable manifold.

Findings

Proves a geometric criterion for first integrals with integrating factors.

Extends results to volume-preserving vector fields with nontrivial normalizers.

Applicable to all smooth orientable manifolds.

Abstract

We prove a sufficient condition for the existence of explicit first integrals for vector fields which admit an integrating factor. This theorem recovers and extends previous results in the literature on the integrability of vector fields which are volume preserving and possess nontrivial normalizers. Our approach is geometric and coordinate-free and hence it works on any smooth orientable manifold.