  In this paper, we discuss the existence of solution (as Cauchy transform of a
signed measure) of a particular algebraic quadratic equation of the form
$\mathcal{C}^{2}\left( z\right) +r\left( z\right) \mathcal{C}\left( z\right)
+s\left( z\right) =0.$ This problem remains to describe the critical graph of a
related polynomial quadratic differential; in particular, we discuss the
existence of finite critical trajectories of this quadratic differential.
