  The modified XY model is a modification of the XY model by addition of a
half-periodic term. The modified Goldstone model is a regular and continuum
version of the modified XY model. The former admits a vortex molecule, that is,
two half-quantized vortices connected by a domain wall, as a regular
topological soliton solution to the equation of motion while the latter admits
it as a singular configuration. Here we define the ${\mathbb Z}_n$ modified XY
and Goldstone models as the $n=2$ case to be the modified XY and Goldstone
models, respectively. We exhaust all stable and metastalble vortex solutions
for $n=2,3$ and find a vortex confinement transition from an integer vortex to
a vortex molecule of $n$ $1/n$-quantized vortices, depending on the ratio
between the term of the XY model and the modified term. We find for the case of
$n=3$, a rod-shaped molecule is the most stable while a Y-shaped molecule is
metastable. We also construct some solutions for the case of $n=4$.The vortex
confinement transition can be understood in terms of the ${\mathbb C}/{\mathbb
Z}_n$ orbifold geometry.
