Payne-Philippin's overdetermined problems on compact surfaces

We investigate the overdetermined problem given by \begin{equation*} Δu=0 $\text{ in }$ Ω,\quad \frac{\partial u}{\partialν} =σ_1 u $\text{ on }$ \partial Ω, \quad |\nabla u|=$\text{constant on }$ \partial Ω, \end{equation*} where $\mathrm{\Omega }$ is a connected compact Riemannian surface with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$, and ${\sigma }_1$ is the first nonzero Steklov eigenvalue of $\mathrm{\Omega }$. We prove that this overdetermined problem admits a nontrivial solution if and only if $\mathrm{\Omega }$ is $\sigma$-homothetic to either the flat unit disk or a flat cylinder $[-T,T]\times S^1$ for some $T\ge T_1$. This gives a complete answer to the question raised by Payne and Philippin in [Z. Angew. Math. Phys. 42(6), 864-873, 1991] for $\sigma ={\sigma }_1$ and arbitrary surfaces. In particular, we completely characterize compact domains in 2-dimensional space forms for which the overdetermined problem is solvable.