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.



Multiplicity of Laplace eigenvalues and symmetry
Let me extend, and correct, the argument expressed in the comment made by user378654.


Let us start with a surface $S$ for which $\Delta$ admits a double eigenvalue $\lambda$. For instance, you may choose an $S$ with a non-trivial symmetry group. Denote $V=\ker(\lambda-\Delta)$. Consider now a smooth $m$-parameter family $p\mapsto S\_p$ around $S$, with say $S\_0=S$. By a Lyapunov-Schmidt procedure, which essentially involves the fact that the spectrum of $\Delta$ is discrete, there exists a unique smooth map $p\mapsto V(p)$ over a neighbourhood $U$ of $0$, where $V(p)$ is a plane in $C^\infty(S\_p)$, such that $V(0)=V$ and $V(p)$ is stable under $\Delta$. Choosing a smoothly varying unitary basis ${\cal B}\_p$ of $V(p)$, the action of $\Delta$ over $V(p)$ is represented by a $2\times2$ symmetric real matrix $M(p)$.


Now if $m=3$ and the family is "generic", in the sense that the map $p\mapsto M(p)$ is a submersion at $p=0$, then the $3$-parameter family $S\_p$ contains a $1$-parameter family of surfaces that admit a double eigenvalue.


Notice the importance of the assumption that $m\ge3$. A one-parameter family $p\mapsto S\_p$ does not suffice. This phenomenon (for symmetric matrices in general) is described in a book of V. I. Arnold.


