  It is shown that Kundt's metric for vacuum cannot be constructed when
two-dimensional space-like sections of null hypersurfaces are compact,
connected manifolds with no boundary unless they are tori or spheres, i.e.
higher genus $\mathbf{g} \geq 2$ is excluded by vacuum Einstein equations. The
so-called {\em basic equation} (resulting from Einstein equations) is examined.
This is a non-linear PDE for unknown covector field and unknown Riemannian
structure on the two-dimensional manifold. It implies several important results
derived in this paper. It arises not only for Kundt's class but also for
degenerate Killing horizons and vacuum degenerate isolated horizons.
