  We investigate a static solution with an hyperbolic nature, characterised by
a pseudo-spherical foliation of space. This space-time metric can be perceived
as an anti-Schwarzschild solution, and exhibits repulsive features. It belongs
to the class of static vacuum solutions termed "a degenerate static solution of
class A". In the present work we review its fundamental features, discuss the
existence of generalised wormholes, and derive its extension to scalar-tensor
gravity theories in general.
