Singular fiber of the Mumford system and rational solutions to the KdV hierarchy

We study the singular isolevel manifold Mg (0) of the genus g Mumford system associated to the spectral curve y² = x^2g+1. We show that Mg (0) is stratified by g +1 open subvarieties of additive algebraic groups of dimension 0,1....g, and we give an explicit description of Mg (0) in terms of the compactification of the generalized Jacobian. As a consequence, we obtain an effective algorithm to compute rational solutions to the genus g Mumford system, which is closely related to rational solutions of the KdV hierarchy.