We consider an Abel equation (*) y' = p(x)y(2) + q(x)y(3) with p(x), q(x) p
olynomials in x. A center condition for (a) (closely related to the classic
al center condition for polynomial vector fields on the plane) is that y(0)
= y(0) = y(1) for any solution y(x) of(*). This condition is given by the
vanishing of all the Taylor coefficients v(k)(1) in the development y(x) =
y(0) + Sigma(k=2)(infinity) v(k)(x)y(0)(k) Anew basis for the ideals I-k =
{v(2),...,v(k)} has recently been produced, defined by a linear recurrence
relation. Studying this recurrence relation, we connect center conditions w
ith a representability of P = integral p and Q = integral q in a certain co
mposition form (developing further some results of Alwash and Lloyd), and w
ith a behavior of the moments integral P-k q. On this base, explicit center
equations are obtained for small degrees of p and q.