Stability of the Thomson vortex polygon with evenly many vortices outside a circular domain

We consider the stability problem for the stationary rotation of a regular point vortex $n$-gon lying outside a circular domain. After the article of Havelock (1931), the complete solution of the problem remains unclear in the case $2\le n\le6$. We obtain the exhaustive results for evenly many vortices $n=2,4,6$.