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Given a foliation F in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case S = P 2 some new counter-examples to the classic formulation of the Poincaré problem are presented. If S is a rational surface and F has singularities of type (1, 1) or (1, −1) we prove that the general solution is a non-singular curve.doi:10.5565/publmat_41297_03 fatcat:6gclmjxrhjehpmcjalvttdjigq