  If on a smooth bounded domain $\Omega\subset\mathbb{R}^2$ there is a
nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary,
must $\Omega$ be a disk or an annulus? This question can be understood as a
weaker analog of the well known Schiffer conjecture, in that the function $u$
is allowed to take a different constant value on each connected component of
$\partial \Omega$ yet many of the known rigidity properties of the original
problem are essentially preserved. Our main result provides a negative answer
by constructing a family of nontrivial doubly connected domains $\Omega$ with
the above property. As a consequence, a certain linear combination of the
indicator functions of the domains $\Omega$ and of the bounded component of the
complement $\mathbb{R}^2\backslash\overline{\Omega}$ fails to have the Pompeiu
property. Furthermore, our construction implies the existence of continuous,
compactly supported stationary weak solutions to the 2D incompressible Euler
equations which are not locally radial.
