Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV

We study a certain family of Schrodinger operators whose eigenfunctions phi(x, lambda) satisfy a differential equation in the spectral parameter lambda of the form B(lambda, partial derivative-lambda)phi = THETA(x)phi. We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class of bispectral potentials. This extends and complements a result of Duistermaat and Grunbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy