  It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace
operator in $\mathbb R^3$ vanishes on a real analytically ruled two-dimensional
surface $S \subset \mathbb R^3$ then $S$ is a union of cones, each of which is
contained in a translate of the zero set of a nonzero harmonic homogeneous
polynomial. If $S$ is an immersed $C^1-$ manifold then $S$ is a Coxeter system
of planes. Full description of common nodal sets of the Laplace spectra of
convexly supported distributions is given. In equivalent terms, the result
describes ruled injectivity sets for the spherical mean transform and confirms,
for the case of ruled surfaces in $\mathbb R^3,$ a conjecture of E.T. Quinto
and the author .
