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

We study a certain family of Schrödinger operators whose eigenfunctions φ{symbol}(χ, λ) satisfy a differential equation in the spectral parameter λ of the form B(λ,∂_λ)φ{symbol}=Θ(x)φ{symbol}. 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 Grünbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy.