Weierstrass integrability of differential equations

The integrability problem consists of finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define Weierstrass integrability and we determine some Weierstrass integrable systems which are Liouvillian integrable. Inside this new class of integrable systems there are non-Liouvillian integrable systems.