Analytic integrability of some examples of degenerate planar vector fields

Abstract

This paper is devoted to the classification of analytic integrable cases of two families of degenerate planar vector fields with a monodromic singular point at the origin. This study falls in the still open degenerate center problem. This classification can be done using the formal normal form theory and knowing a suitable normal form of any differential systems associated to each family.

The authors are grateful to the referees for their valuable remarks which helped to improve the manuscript. The first and second authors are supported by a MINECO/FEDER grant number MTM2010-20907-C02-02 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (projects P12-FQM-1658; TIC-130 and FQM-276). The third author is partially supported by a MINECO/FEDER grant number MTM2011- 22877 and by an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204.