### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 035, 9 pages      arXiv:1206.3005      http://dx.doi.org/10.3842/SIGMA.2012.035

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

Jaume Llibre a and Daniel Peralta-Salas b
a) Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
b) Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain

Received February 16, 2012, in final form June 12, 2012; Published online June 14, 2012

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.

Key words: first integral; vector field; integrating factor; normalizer.

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