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In this paper, we discuss the existence of solution (as Cauchy transform of asigned 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 arelated polynomial quadratic differential; in particular, we discuss theexistence of finite critical trajectories of this quadratic differential.

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