Solution of the center problem for polynomial Abel equations

We prove that, if the Poincar\'{e} map $y(0)\longmapsto y(1)$ for solutions $y(x)$ of the polynomial Abel equation $dy/dx=P^{\prime}(x) y^{2} + Q^{\prime}(x) y^{3}$ is the identity, then the polynomials $P$ and $Q$ are compositions with a nonconstant polynomial $R(x)$ such that $R(0)=R(1).$