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

Spectral curves in gauge/string dualities: integrability, singular sectors and regularization