Center and Moment Conditions for Rational Abel Equation

We consider the Abel Equation dy dz = p(z)y2 + q(z)y3 as an equation on the complex plane with p, q – rational functions. The center problem for this equation (which is closely related to the classical center problem for polynomial vector fields on the plane) is to find conditions on p and q under which all the solutions y(z) are univalued functions along the circle |z| = 1. In [3] we have shown that this problem is closely related to Moment Problem, namely conditions on p, q for vanishing of all the moments ∫ |z|=1 P iQjdP with P = ∫ p, Q = ∫ q. We slightly generalize an approach and consider an arbitrary curve γ ∈ C in place of the unit circle |z| = 1. The aim of this paper is to give a simple and constructive description of rational functions P and Q satisfying Moment Condition along γ, and to show that Moment Condition implies Center Condition for P and Q – Laurent polynomials.