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

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