Belyi pairs corresponding to dessins d'enfants of genus $3$

In this article we discuss a question arising in the dessins d'enfants theory: we are interested in finding the defining equations for the Belyi pair of a given dessin. Low genus ($g\le2$) cases are well-studied, there exist many examples of Belyi pairs of those genera and some effective methods have been developed. We present a method which allows to find Belyi pairs of self-dual genus $3$ dessins.