  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.
