  We define the notion of an exceptional manifold to be a flat Riemannian
manifold with boundary which supports a positive harmonic function satisfying
simultaneously a zero Dirichlet condition and a constant (nonzero) Neumann
condtion at the boundary. We study the two-dimensional case: we present various
examples and give a general construction algorithm of such surface by using
complex analysis. We deduce a classification of all such surfaces assuming some
further natural hypotheses and prove a Bernstein type theorem.
