  Using Roelcke formula for the Green function, we explicitly construct a basis
in the kernel of the adjoint Laplacian on a compact polyhedral surface $X$ and
compute the $S$-matrix of $X$ at the zero value of the spectral parameter. We
apply these results to study various self-adjoint extensions of a symmetric
Laplacian on a compact polyhedral surface of genus two with a single conical
point. It turns out that the behaviour of the $S$-matrix at the zero value of
the spectral parameter is sensitive to the geometry of the polyhedron.
