Nonintegrability of a class of the Bianchi V I<inf>0</inf> and V II<inf>0</inf> models

Jaume Llibre, Clàudia Valls

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1 Citation (Scopus)


The well-known Bianchi V I0 and Bianchi V II0 dynamical systems are three-dimensional differential systems which after a convenient reduction become over(x, ̇) = - x2 + (z + 1) y2, over(y, ̇) = - 4 (z + 1) + x y z, over(z, ̇) = - y z (z + 2) . In the paper of Maciejewski and Szydiowski [A.J. Maciejewski, M. Szydiowski, Bianchi cosmologies as dynamical systems, Celestial Mech. Dynam. Astronom. 73 (1999) 17-24], the authors asked about the integrability or nonintegrability of this system. Here we show that this system has no first integrals which are polynomial, rational, Darboux functions or analytic functions. Consequently this system is not integrable inside these classes of functions. © 2010 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)815-822
JournalJournal of Geometry and Physics
Issue number5
Publication statusPublished - 1 May 2010


  • Analytic first integrals
  • Bianchi V I model 0
  • Bianchi V II model 0
  • Darboux first integrals
  • Darboux polynomials
  • Formal first integrals
  • Rational first integrals

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