On the integrability of two-dimensional flows

This paper deals with the notion of integrability of flows or vector fields on two-dimensional manifolds. We consider the following two key points about first integrals: (1) They must be defined on the whole domain of definition of the flow or vector field, or defined on the complement of some special orbits or the system; (2) How are they computed? We prove that every local flow phi on a two-dimensional manifold M always has a continuous first integral on each component of M\Sigma where Sigma is the set of all separatrices of phi. We consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two-dimensional flow or vector field. Finally, we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus, of a non-resonant hyperbolic node, and of a Siegel hyperbolic saddle. (C) 1999 Academic Press.