Vortex Pairs on the Triaxial Ellipsoid: Axis Equilibria Stability

We consider a pair of opposite vortices moving on the surface of the triaxial ellipsoid $\mathbb{E}(a,b,c): \, x^2/a + y^2/b + z^2/c = 1,\, a<b<c$. The equations of motion are transported to $S^2 \times S^2$ via a conformal map that combines confocal quadric coordinates for the ellipsoid and sphero-conical coordinates in the sphere. The antipodal pairs form an invariant submanifold for the dynamics. We characterize the linear stability of the equilibrium pairs at the three axis endpoints.