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

We study a certain family of Schroedinger operators whose eigenfunctions φ(x, λ) satisfy a differential equation in the spectral parameter λ of the form B(λ, dλ)φ=Θ(x)φ. 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 Gruenbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy. (orig.)