On the polynomial differential systems having polynomial first integrals

Belén García, Jaume Llibre, Jesús S. Pérez del Río

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We consider the class of complex planar polynomial differential systems having a polynomial first integral. Inside this class the systems having minimal polynomial first integrals without critical remarkable values are the Hamiltonian ones. Here we mainly study the subclass of polynomial differential systems such that their minimal polynomial first integrals have a unique critical remarkable value. In particular we characterize all the Liénard polynomial differential systems x=y, y=-f(x)y-g(x), with f(x) and g(x) complex polynomials in the variable x, having a minimal polynomial first integral with a unique critical remarkable value. © 2011 Elsevier Masson SAS.
Original languageEnglish
Pages (from-to)309-316
JournalBulletin des Sciences Mathematiques
Issue number3
Publication statusPublished - 1 Apr 2012


  • Critical remarkable value
  • Liénard differential system
  • Polynomial differential system
  • Polynomial first integral

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