Center conditions II: 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 relat ed to the classical center condition for polynomial vector fields on the pl ane) is that y(0) = y(0) equivalent to y(1) for any solution y(x) of (*). We introduce 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 cent er, if for any epsilon and for any solution y(epsilon, x) of (**), y(epsilo n, 0) equivalent to y(epsilon, 1). We show that the parametric center condition implies vanishing of all the m oments m(k)(1), where m(k)(x) = integral(0)(x) P-k(t)q(t)dt, P(x) = integra l(0)(x)p(t)dt. We investigate the structure of zeroes of m(k)(x) and on thi s base prove in some special cases a composition conjecture, stated in [10] , for a parametric center problem.