Foliations in algebraic surfaces having a rational first integral

Given a foliation $\Cal F$ in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case $S=\Bbb P^2$ some new counter-examples to the classic formulation of the Poincaré problem are presented. If $S$ is a rational surface and $\Cal F$ has singularities of type $(1,1)$ or $(1,-1)$ we prove that the general solution is a non-singular curve.