  This paper deals with the determination of the S-curves in the theory of
non-hermitian orthogonal polynomials with respect to exponential weights along
suitable paths in the complex plane. It is known that the corresponding complex
equilibrium potential can be written as a combination of Abelian integrals on a
suitable Riemann surface whose branch points can be taken as the main
parameters of the problem. Equations for these branch points can be written in
terms of periods of Abelian differentials and are known in several equivalent
forms. We select one of these forms and use a combination of analytic an
numerical methods to investigate the phase structure of asymptotic zero
densities of orthogonal polynomials and of asymptotic eigenvalue densities of
random matrix models. As an application we give a complete description of the
phases and critical processes of the standard cubic model.
