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.