A note on Liouvillian first integrals and invariant algebraic curves

In this work we study the existence and non-existence of finite invariant algebraic curves for a complex planar polynomial differential system having a Liouvillian first integral.The first and the second authors are partially supported by a MICINN /FEDER grant number MTM2011-22877 and by a AGAUR (Generalitat de Catalunya) grant number 2009SGR 381. The third author is partially supported by a MICINN/FEDER grant num- ber MTM2008-03437, by a AGAUR grant number 2009SGR 410 and by ICREA Academia