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) = y(1) for any solution y(x) of (*).
Following [7], we consider a parametric version of this condition: an equat
ion
(**) 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 cen
ter, if for any epsilon and for any solution y(epsilon, x) of (**) y(epsilo
n, 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) = in
tegral(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.