Solution of the parametric center problem for the Abel differential equation

The Abel differential equation y' = p(x)y² + q(x)y³ with p, q ∈ ℝ[x] is said to have a center on an interval [a, b] if all its solutions with the initial value y(a) small enough satisfy the condition y(b) = y(a). The problem of description of conditions implying that the Abel equation has a center may be interpreted as a simplified version of the classical center-focus problem of Poincaré. The Abel equation is said to have a "parametric center" if for each ϵ ∈ ℝ the equation y' = p(x)y² +ϵq(x)y³ has a center. In this paper we show that the Abel equation has a parametric center if and only if the antiderivatives P = ∫ p(x) dx, Q = ∫ q(x) dx satisfy the equalities P = P ○ W, Q = Q ○ W for some polynomials P, Q, and W such that W(a) = W(b). We also show that the last condition is necessary and sufficient for the "generalized moments" ∫^b_a Pⁱ dQ and ∫^b_a QⁱdP to vanish for all i ≥ 0.