About Integrability of the Degenerate System

  • Victor F. EdneralEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


We study integrability of an autonomous planar polynomial system of ODEs with a degenerate singular point at the origin depending on five parameters. By mean of the Power Geometry Method, this degenerated system is reduced to a non-degenerate form by the blow-up process. After, we search for the necessary conditions of local integrability by the normal form method. We look for the set of necessary conditions on parameters under which the original system is locally integrable near the degenerate stationary point. We found seven two-parametric families in the five-parameter space. Then first integrals of motion were found for six families. For the seventh family, we found the formal first integral. So, at least six of these families in parameters space are manifolds where the global integrability of the original system takes place.


Ordinary differential equations Integrability Resonant normal form Power geometry Computer algebra 



The author is very grateful to Profs. A.D. Bruno, V.G. Romanovski, and A.B. Batkhin for important advices, discussions, and assistance.


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Authors and Affiliations

  1. 1.Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State UniversityMoscowRussian Federation
  2. 2.Peoples’ Friendship University of Russia (RUDN University)MoscowRussian Federation

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