  We consider the eigenvalue problem $\Delta^{\mathbb{S}^2} \xi + 2 \xi=0 $ in
$ \Omega $ and $\xi = 0 $ along $ \partial \Omega $, being $\Omega$ the
complement of a disjoint and finite union of smooth and bounded simply
connected regions in the two-sphere $\mathbb{S}^2$. Imposing that $|\nabla
\xi|$ is locally constant along $\partial \Omega$ and that $\xi$ has infinitely
many maximum points, we are able to classify positive solutions as the
rotationally symmetric ones. As a consequence, we obtain a characterization of
the critical catenoid as the only embedded free boundary minimal annulus in the
unit ball whose support function has infinitely many critical points.
