Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane

In this note we construct smooth bounded domains $\Omega \subset \mathbb R^2$, other than disks, for which the overdetermined problem $$\left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b &\qquad& \text{ on } \partial \Omega, \newline \frac{\partial u}{\partial n} &= c &\qquad& \text{ on } \partial \Omega \end{alignedat} \right.$$ has a solution for some constants $\lambda,b,c \ne 0$. These appear to be the first counterexamples to a conjecture of Willms and Gladwell [WG94].