Integrability of complex planar systems with homogeneous nonlinearities

Abstract

In this paper we obtain sufficient conditions for the existence of a local analytic first integral for a family of quintic systems having homogeneous nonlinearities. The family studied in this work is the largest one classified until now for systems with such nonlinearities. We propose also an approach to find reversible systems within polynomial families of Lotka-Volterra systems with homogeneous nonlinearities.

Brigita Fercec and Valery Romanovski acknowledge the support of the study by the Slovenian Research Agency and by a Curie International Research Staff Exchange SchemeFellowship within the 7th European Community Framework Programme, FP7-PEOPLE-2012- IRSES-316338. Jaume Giné is partially supported by a MINECO/ FEDER grant number MTM2014-53703-P and by an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. Victor Edneral acknowledges the support by the Russian Federation grant on support of scientific schools number 3042.2014.2.