Berenstein, Carlos A.Yans, P.C.We consider the problem of deciding whether the over-determined Neumann eigenvalue boundary value problem: *DELTA*u + ALPHA*u = 0 in D; u = 1, SOME GREEK SYMBOL*u/SOME GREEK SYMBOL*n = 0 on SOME GREEK SYMBOL*D 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.en-USTechnical Report