We establish precise spectral criteria for potential functions V of reflectionless Schrödinger operators L_V = −∂x² + V to admit solutions to the Korteweg–de Vries (KdV) hierarchy with V as an initial value. More gener-ally, 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.