A solution to the Pompeiu problem

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of distributions, and $$\int_{\sigma (D)} f(x) dx = 0 \quad \forall \sigma \in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain, the closure $\bar{D}$ of which is diffeomorphic to a closed ball, and $S$ is its boundary. Then the comp$is connected and path connected. By$G$the group of all rigid motions of$\R^n is denoted. This group consists of all translations and rotations. A proof of the following theorem is given. Theorem 1. {\it Assume that n=2$,$f\not\equiv 0$, and (*) holds. Then$D is a ball.} Corollary. {\it If the problem (\nabla^2+k^2)u=0$in$D$,$u_N|_S=0$,$u|_S=const\neq 0$has a solution, then$D is a ball.} Here N$is the outer unit normal to$S$.