Polynomial first integrals via the Poincaré series

In (Appl. Math. 28 (2001) 17) a method to compute the Poincaré–Liapunov constants for an arbitrary analytic differential system which has a linear center at the origin in function of the coefficients of the system was given. This method also computes the coefficients of the Poincaré series in function of the same coefficients. In this work we use this method to determine polynomial differential systems which have a polynomial first integral.