We consider an Abel equation (*)y’=p(x)y2 +q(x)y3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty0=y(0)≡y(1) for any solutiony(x) of (*).
Folowing , we consider a parametric version of this condition: an equation (**)y’=p(x)y2 +εq(x)y3p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1)..
We give another proof of the fact, shown in , that the parametric center condition implies vanishing of all the momentsmk (1), wheremk(x)=∫0xpk(t)q(t)(dt),P(x)=∫0xp(t)dt. We investigate the structure of zeroes ofmk(x) and generalize a “canonical representation” ofmk(x) given in . On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem.