Nonintegrabilty of a Halphen system

J. Llibre^a, C. Valls<sup>b

a</sup> Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Catalonia, Spain
^b Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Lisboa, Portugal

Abstract: We study the Halphen system with real variables and real constants. We show that in the case where at least one constant is nonzero, this system does not admit any first integral that can be described by formal power series. It hence follows that analytic first integrals do not exist. Furthermore, we prove that first integrals of the Darboux type also do not exist.

Keywords: Halphen system, analytic first integral.

MSC: 34C35, 34D30

Received: 05.05.2014

DOI: 10.4213/tmf8705

 English version: Theoretical and Mathematical Physics, 2014, 181:2, 1383–1395

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