In order to investigate polynomial vector ¯elds admitting a prescribed Darboux integrating factor, we show that it is helpful to employ morphisms of the a±ne plane. In particular, such morphisms may be used to remove degeneracies of the underlying curve. Our main result states that the space of vector ¯elds admitting a prescribed Darboux integrating factor modulo a well understood subspace has ¯nite dimension. This extends earlier work for the case of nondegenerate geometric setting. In addition, we present a number of explicit examples with degenerate underlying curve.