  We discuss the nature of criticality in the $\beta^2 = 2 \pi N$ self-dual
extention of the sine-Gordon model. This field theory is related to the
two-dimensional classical XY model with a N-fold degenerate symmetry-breaking
field. We briefly overview the already studied cases $N=2,4$ and analyze in
detail the case N=3 where a single phase transition in the three-state Potts
universality class is expected to occur. The Z$_3$ infrared critical properties
of the $\beta^2 = 6 \pi$ self-dual sine-Gordon model are derived using two
non-perturbative approaches. On one hand, we map the model onto an integrable
deformation of the Z$_4$ parafermion theory. The latter is known to flow to a
massless Z$_3$ infrared fixed point. Another route is based on the connection
with a chirally asymmetric, su(2)$_4$ $\otimes$ su(2)$_1$
Wess-Zumino-Novikov-Witten model with anisotropic current-current interaction,
where we explore the existence of a decoupling (Toulouse) point.
