On the Formal Integrability Problem for Planar Differential Systems

We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results obtained, we consider some families of analytic differential systems in C2, and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.A.Algaba andC. Garc´ıa are supported by aMICINN/FEDER Grant no. MTM2010-20907-C02-02 and by the Consejer´ıa de Educac´ıon y Ciencia de la Junta de Andaluc´ıa (projects EXC/2008/FQM-872, TIC-130, and FQM-276). J. Gin´e is partially supported by a MICINN/FEDER Grant no. MTM2011- 22877 and by Generalitat de Catalunya Grant no. 2009SGR 381