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In this paper, we discuss the representability almost everywhere (a.e.) inthe plane of an irreducible algebraic function as the Cauchy transform of asigned measure supported on a finite number of compact semi-analytic curves anda finite number of isolated points. We discuss the existence of criticaltrajectories of a family of quadratic differentials.

Quadratic differentials A(z-a)(z-b)dz^2/(z-c)^2 and algebraic Cauchy transform