  We prove that foliations on the projective plane admitting a Liouvillian
first integral but not admitting a rational first integral always have
invariant algebraic curves of degree bounded by a function of the degree of the
foliation. We establish, for the same class of foliations, the existence of a
bound for the degree of the simplest integrating factor depending only on the
degree of the foliation and on the nature of its singularities. We also prove
the existence of invariant algebraic curves of small degree for foliations with
rational first integral and intermediate Kodaira dimension.
