Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in ℝ³

It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in R3 vanishes on a real-analytically ruled two-dimensional surface S R³ 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 C1 manifold then S is a Coxeter system of planes. Full description of common nodal sets of 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 R3; a conjecture from [1].