Solution of the Hurwitz problem for Laurent polynomials

In this paper we investigate the following existence problem for rational functions: for a given collection $\Pi$ of partitions of a number $n$ to define whether there exists a rational function $f$ of degree $n$ for which $\Pi$ is the branch datum. An important particular case when the answer to this problem is known is the one when the collection $\Pi$ 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 $\Pi$ contains a partition consisting of two elements.