KdV hierarchy via Abelian coverings and operator identities

We establish precise spectral criteria for potential functions $ V$ of reflectionless Schrödinger operators $ L_V = -\partial _x^2 + V$ to admit solutions to the Korteweg-de Vries (KdV) hierarchy with $ V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker-Akhiezer function on infinitely connected domains with a uniformly thick boundary satisfying a fractional moment condition.