We consider the problem of deciding whether the over-determined Neumann eigenvalue boundary value problem: DELTAu + ALPHAu = 0 in D; u = 1, SOME GREEK SYMBOLu/SOME GREEK SYMBOLn = 0 on SOME GREEK SYMBOLD has a solution. This problem arises in thermodynamics and in harmonic analysis. We show that the existence of infinitely many solutions is equivalent to D being a ball.