Ref. [@Hartle:1972ya] in the context of Israel-Wilson-Perjés (IWP) solutions of the Einstein-Maxwell theory. As shown by Tod in Ref. [@Tod:1983pm] these are precisely the timelike solutions of pure $N=2,d=4$ supergravity and a special case of the general problem that we are going to study. Hartle and Hawking were motivated, not by supersymmetry, but rather by the prospect of finding regular solutions describing more than one black hole. They were, in particular, worried about possible string singularities related to NUT charges. These singularities can be eliminated by compactifying the time coordinate with certain period [@kn:M], but at the price of losing asymptotic flatness. Let us consider a possible string singularity parametrized by $z$ and choose polar coordinates $\rho,\phi$ around it. If one considers the integral of the 1-form $\omega$ that appears in the metric along a loop of radius $R$ enclosing the possible string singularity at two different points $z_{1}$ and $z_{2}$, denoted by $I(R,z_{1,2})$, one can use Stokes’ theorem to derive

$$I(R,z_{1})-I(R,z_{2}) = \int_{\Sigma^{2}} d\omega = 2\int_{\Sigma^{2}}
\star_{3}\Im{\rm m}\mathcal{W}\, ,$$

where $\Sigma^{2}$ is a surfaces whose boundaries are the loops of radius $R$ at $z_{1,2}$. In the zero radius limit $\Sigma^{2}$ is a closed surface that crosses the possible string 