singularity at $z_{1}$ and $z_{2}$ and we have

$$2\pi \lim_{R\rightarrow 0} R [\omega_{\phi}(R,z_{1})-\omega_{\phi}(R,z_{2})]
= 2\int_{\Sigma^{2}}
\star_{3}\Im{\rm m}\mathcal{W}= \int_{\Sigma^{3}}
d\star_{3}\Im{\rm m}\mathcal{W}= 2\int_{\Sigma^{3}} d^{3}x 
\Im{\rm m}\, (\partial_{\underline{m}}\mathcal{W}_{\underline{m}})\, ,$$

where $\partial\Sigma^{3}=\Sigma^{2}$. Thus, $\Im{\rm m}\,
(\partial_{\underline{m}}\mathcal{W}_{\underline{m}})\neq 0$ implies that $\omega_{\phi}$ is singular on the string somewhere between $z_{1}$ and $z_{2}$. These singularities are related to the presence of NUT sources, since we can define the NUT charge contained in $\Sigma^{3}$ as the integral of $d\omega$ over $\Sigma^{2}=\partial\Sigma^{3}$:

$$-8\pi N_{\Sigma}= \int_{\Sigma^{2}}d\omega = \int_{\Sigma^{3}}d^{2}\omega = 
2\int_{\Sigma^{3}} d^{3}x 
\Im{\rm m}\, (\partial_{\underline{m}}\mathcal{W}_{\underline{m}})\, .$$

Thus, the condition $\Im{\rm m}\,
(\partial_{\underline{m}}\mathcal{W}_{\underline{m}})=0$, required by supersymmetry, is equivalent to the absence of sources of NUT charge.

Hartle and Hawking argued that the only solutions in the IWP class with no NUT charge (and no singularities) were the Majumdar-Papapetrou solutions [@Majumdar:1947eu; @kn:Pa] which are regular and static. We will review their arguments in Section \[sec-twobhsouren2sugra\] and show that there are indeed non-trivial solutions that satisfy the KSIs and have no NUT charges, apart from the Majumdar-Papapetrou ones; they all have negative total mass, which causes other naked singularities to appear.

Thus, if we include positivity of all masses among the requirements necessary to have supersymmetry, the only supersymmetric black-holes-type solutions of pure $N=2,d=4$ 