Solutions of the Hurwitz problem for laurent polynomials

We investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer is known is the one when the collection Π contains a partition consisting of a single element (in this case, the corresponding rational function is equivalent to a polynomial). In this paper, we provide a solution in the case when Π contains a partition consisting of two elements.