A new integrable problem of motion of point vortices on the sphere

The dynamics of an antipodal vortex on a sphere (a point vortex plus its antipode with opposite circulation) is considered. It is shown that the system of $n$ antipodal vortices can be reduced by four dimensions (two degrees of freedom). The cases $n=2,3$ are explored in greater detail both analytically and numerically. We discuss Thomson, collinear and isosceles configurations of antipodal vortices and study their bifurcations.