Trajectories of a quadratic differential related to a quasi-exactly solvable sextic oscillator

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.