Rational solutions of the master symmetries of the KdV equation

In this article we characterize a certain class of rational solutions of th e hierarchy of master symmetries for KdV. The result is that the generic ra tional potentials that decay at infinity and remain rational by all the flo ws of the master-symmetry KdV hierarchy are bispectral potentials for the S chrodinger operator. By bispectral potentials we mean that the correspondin g Schrodinger operators possess families of eigenfunctions that are also ei genfunctions of a differential operator in the spectral variable. This comp lements certain results of Airault-McKean-Moser [4], Duistermaat-Grunbaum [ 10], and Magri-Zubelli [40]. As a consequence of bispectrality, the rationa l solutions of the master symmetries turn out to be solutions of a (general ized) string equation.