On New Integrals of the Algaba-Gamero-Garcia System

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10490)


We study local integrability of a plane autonomous polynomial system of ODEs depending on five parameters with a degenerate singular point at the origin. The approach is based on making use of the Power Geometry Method and the computation of normal forms. We look for the complete set of necessary conditions on parameters of the system under which the system is locally integrable near the degenerate stationary point. We found earlier that the sets of parameters satisfying these conditions consist of four two-parameter subsets in the full five-parameter co-space. Now we consider the special subcase of the case \(b^2 = 2/3\) and separate subsubcases when additional first integrals can exist. Here we have found two such integrals.


Ordinary differential equations Integrability Resonant normal form Power Geometry Computer algebra 



Victor F. Edneral was supported by the grant NSh-7989.2016.2 of the President of Russian Federation and by the Ministry of Education and Science of the Russian Federation (Agreement number 02 A03.21.0008), Valery G. Romanovski was supported by the Slovenian Research Agency (research core funding No. P1-0306).


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Keldysh Institute of Applied Mathematics of RASMoscowRussia
  2. 2.Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State UniversityMoscowRussia
  3. 3.Peoples’ Friendship University of Russia (RUDN University)MoscowRussian Federation
  4. 4.Faculty of Electrical Engineering and Computer ScienceUniversity of MariborMariborSlovenia
  5. 5.CAMTP – Center for Applied Mathematics and Theoretical PhysicsUniversity of MariborMariborSlovenia
  6. 6.Faculty of Natural Science and MathematicsUniversity of MariborMariborSlovenia

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