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We establish precise spectral criteria for potential functions $V$ ofreflectionless Schr\"odinger operators $L_V = -\partial_x^2 + V$ to admitsolutions to the Korteweg de-Vries (KdV) hierarchy with $V$ as an initialvalue. More generally, our methods extend the classical study ofalgebro-geometric solutions for the KdV hierarchy to noncompact Riemannsurfaces by defining generalized Abelian integrals and analogues of theBaker-Akhiezer function on infinitely connected domains with a uniformly thickboundary satisfying a fractional moment condition.

KdV hierarchy via Abelian coverings and operator identities