Let $S$ be a smooth closed connected hyperbolic surface. On $S$ we have the Laplace operator $\Delta$, whose eigenvalues form a discrete sequence \begin{equation} 0=\lambda_0<\lambda_1\leq \lambda_2\leq ... \leq \lambda_k\nearrow\infty \end{equation} To my knowledge, a generic surface will not have eigenvalues of multiplicity $>1$ (this is a result of K.Uhlenbeck, see https://www.jstor.org/stable/2374041). On the other hand, any surface with degenerate eigenvalues (such as the Bolza surface or Klein quartic), which I am aware of, usually has a lot of symmetries (in the sense that the automorphism group of $S$ is large), which cause the high multiplicities. Are there any known inverse results of the type "If a hyperbolic surface $S$ has an eigenvalue of multiplicity >1, then it has non-trivial automorphism group". I was unable to spot any related results in the literature, so I want to to ask whether there is anything known in this direction or if there are any examples of hyperbolic surfaces violating this idea.