On the integrability of two-dimensional flows

Javier Chavarriga, Hector Giacomini, Jaume Giné, Jaume Llibre

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110 Citations (Scopus)


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 of the system; (2) How are they computed? We prove that every local flow φ on a two-dimensional manifold M always has a continuous first integral on each component of M\Σ where Σ is the set of all separatrices of φ. 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. © 1999 Academic Press.
Original languageEnglish
Pages (from-to)163-182
JournalJournal of Differential Equations
Publication statusPublished - 1 Sept 1999


  • First integral
  • Inverse integrating factor
  • Two-dimensional differential systems


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