Abstract
In this paper, we discuss the existence of a solution interpreted as the Cauchy transform of a signed measure of a particular algebraic quadratic equation of the form zC2(z) - P (z) C(z) + Q (z) = 0 for some polynomials P (z) and Q (z) . This issue requires the description of the critical graph of a related quadratic differential in the Riemann sphere C. In particular, we discuss the existence of finite critical trajectories of this quadratic differen-tial.