Center conditions III: Parametric and model center problems

We consider an Abel equation (*) y' = p(X)y(2) + q(x)y(3) with p(x), q(x) polynomials in x. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is that y(0) = y(0) = y(1) for any solution y(x) of (*). Following [7], we consider a parametric version of this condition: an equation (**) y' = p(x)y(2) + epsilon q(x)y(3), p, q as above, epsilon is an element of C, is said to have a parametric center, if for any epsilon and for any solution y(epsilon, x) of (**) y(epsilon, 0) = (epsilon, 1). We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the moments m(k)(1), where m(k)(x) = integral(0)(x) P-k(t)q(t)dt, P(x) = integral(0)(x) p(t)dt. We investigate the structure of zeroes of m(k)(x) and generalize a "canonical representation" of m(k)(x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem.