  We study the two-dimensional generalized XY model that depends on an integer
$q$ by the Monte Carlo method. This model was recently proposed by Romano and
Zagrebnov. We find a single Kosterlitz-Thouless (KT) transition for all values
of $q$, in contrast with the previous speculation that there may be two
transitions, one a regular KT transition and another a first-order transition
at a higher temperature. We show the universality of the KT transitions by
comparing the universal finite-size scaling behaviors at different values of
$q$ without assuming a specific universal form in terms of the KT transition
temperature $T_{\rm KT}$.
