  We study a generalization of the XY model with an additional nematic-like
term through extensive numerical simulations and finite-size techniques, both
in two and three dimensions. While the original model favors local alignment,
the extra term induces angles of $2\pi/q$ between neighboring spins. We focus
here on the $q=8$ case (while presenting new results for other values of $q$ as
well) whose phase diagram is much richer than the well known $q=2$ case. In
particular, the model presents not only continuous, standard transitions
between Berezinskii-Kosterlitz-Thouless (BKT) phases as in $q=2$, but also
infinite order transitions involving intermediate, competition driven phases
absent for $q=2$ and 3. Besides presenting multiple transitions, our results
show that having vortices decoupling at a transition is not a suficient
condition for it to be of BKT type.
