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Analytic integrability for some degenerate planar vector fields

Authors:	Antonio Algaba Cristóbal García Jaume Giné

Institution:	1. Departamento de Matemáticas, Centro de Investigación de Física Teórica y Matemática FIMAT, Universidad de Huelva, 21071 Huelva, Spain;2. Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II 69, 25001 Lleida, Catalonia, Spain

Abstract:	In this paper we study the analytic integrability of degenerate vector fields of the form (y³+2ax³y+?,−x⁵−3ax²y²+?) $(y^{3} + 2 a x^{3} y + ?, - x^{5} - 3 a x^{2} y^{2} + ?)$ around the origin. For these vector fields it is proved that integrability does not imply formal orbital equivalence to the Hamiltonian leading part. Moreover, it is shown the existence of a system in this class which has a center but is neither analytically integrable nor formal orbital reversible.

Keywords:	primary 34C05 secondary 34C23 37G15
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