  The singular sector of zero genus case for the Hermitian random matrix model
in the large N limit is analyzed. It is proved that the singular sector of the
hodograph solutions for the underlying dispersionless Toda hierarchy and the
singular sector of the 1-layer Benney (classical long wave equation) hierarchy
are deeply connected. This property is due to the fact that the hodograph
equations for both hierarchies describe the critical points of solutions of
Euler-Poisson-Darboux equations E(a,a), with a=-1/2 for the dToda hierarchy and
a=1/2 for the 1-layer Benney hierarchy.
