  We consider the Whitham equations for deformations of hyperelliptic spectral
curves, which preserve all periods of a meromorphic differential. If the
meromorphic differential has a root at a fixed point of the hyperelliptic
involution, then the Whitham flow has a singularity. We prove that the stable
and unstable manifolds are non-empty and extend the Whitham flow continuously
through the singularity.
