  We study the moduli space of the spectral curves $y^2=W'(z)^2+f(z)$ which
characterize the vacua of $\mathcal{N}=1$ U(n) supersymmetric gauge theories
with an adjoint Higgs field and a polynomial tree level potential $W(z)$. It is
shown that there is a direct way to associate a spectral density and a
prepotential functional to these spectral curves. The integrable structure of
the Whitham equations is used to determine the spectral curves from their
moduli. An alternative characterization of the spectral curves in terms of
critical points of a family of polynomial solutions $\mathbb{W}$ to
Euler-Poisson-Darboux equations is provided. The equations for these critical
points are a generalization of the planar limit equations for one-cut random
matrix models. Moreover, singular spectral curves with higher order branch
points turn out to be described by degenerate critical points of $\mathbb{W}$.
As a consequence we propose a multiple scaling limit method of regularization
and show that, in the simplest cases, it leads to the Painlev\`{e}-I equation
and its multi-component generalizations.
