  We consider a two-dimensional system with two order parameters, one with O(2)
symmetry and one with O($M$), near a point in parameter space where they couple
to become a single O($2+M$) order. While the O(2) sector supports vortex
excitations, these vortices must somehow disappear as the high symmetry point
is approached. We develop a variational argument which shows that the size of
the vortex cores diverges as $1/\sqrt{\Delta}$ and the
Berezinskii-Kosterlitz-Thouless transition temperature of the O(2) order
vanishes as $1/\ln(1/\Delta)$, where $\Delta$ denotes the distance from the
high-symmetry point. Our physical picture is confirmed by a renormalization
group analysis which gives further logarithmic corrections, and demonstrates
full symmetry restoration within the cores.
