  In [arXiv:2204.13980], we recently proposed a modification of general
relativity in which a non-dynamical term related to topology is introduced in
the Einstein equation. The original motivation for this theory is to allow for
the non-relativistic limit to exist in any physical spacetime topology. In the
present paper, we derive an exact static vacuum spherically symmetric solution
of this theory. The metric represents a black hole (with positive Komar mass)
and a naked white hole (with negative and opposite Komar mass) at opposite
poles of an $\mathbb{S}^3$ universe. The solution is similar to the
Schwarzschild metric, but the spacelike infinity is cut, and replaced by a
naked white hole at finite distance, implying that the spacelike hypersurfaces
of the Penrose--Carter diagram are closed. This solution further confirms a
result of [arXiv:2212.00675] suggesting that staticity of closed-space models
in this theory requires a vanishing total mass. As a subcase of the solution,
we also obtain a vacuum homogeneous 3-sphere, something not possible in general
relativity. We also put in perspective the solution with other attempts at
describing singularities on $\mathbb{S}^3$ and discuss how this theory could be
used to study the behaviour of a black hole in an empty closed expanding
universe.
