Let $(M, g)$ be a compact Riemannian $3$-manifold with boundary. Is it possible to exist two independent Neumann eigenfunctions $u, v \in C^{\infty}(M)$ associated to the same eigenvalue $\mu > 0$ of the Laplacian $\Delta_g$ and such that $\nabla u(p)$ and $\nabla v(p)$ are parallel for any $p \in \partial M$? It looks a rather strong condition to impose. Are there any examples?
This showed up in my research, so I appreciate any ideas or comments.