Characterization of the Riccati and Abel Polynomial Differential Systems Having Invariant Algebraic Curves

The Riccati polynomial differential systems are differential systems of the form $x^{\prime }=c_0(x)$, $y^{\prime }=b_0(x)+b_1(x)y+b_2(x)y^2$, where $c_0$ and $b_i$ for $i=0,1,2$ are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable $y$ defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system.