  We consider the question whether a static potential on an asymptotically flat
3-manifold can have nonempty zero set which extends to the infinity. We prove
that this does not occur if the metric is asymptotically Schwarzschild with
nonzero mass. If the asymptotic assumption is relaxed to the usual assumption
under which the total mass is defined, we prove that the static potential is
unique up to scaling unless the manifold is flat. We also provide some
discussion concerning the rigidity of complete asymptotically flat 3-manifolds
without boundary that admit a static potential.
