  We establish precise spectral criteria for potential functions $V$ of
reflectionless Schr\"odinger 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.
