Liouvillian Integrability Versus Darboux Polynomials

Jaume Llibre, Claudia Valls, Xiang Zhang

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3 Citations (Scopus)


© 2016, Springer International Publishing. In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system x˙=y,y˙=-f(x)y-g(x) with deg f> deg g is not Liouvillian integrable.
Original languageEnglish
Pages (from-to)503-515
JournalQualitative Theory of Dynamical Systems
Issue number2
Publication statusPublished - 1 Oct 2016


  • Darboux Jacobian multiplier
  • Darboux polynomial
  • Liouvillian integrability
  • Polynomial differential system


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