Kerr--NUT--Levi-Civita geometries from Ernst inversion: axis structure, curvature singularities, and the Manko--Ruiz parameter

We construct and analyze stationary, axisymmetric vacuum metrics obtained by magnetic Ernst inversion of Kerr-NUT with Manko-Ruiz parameter $C$ and pre-inversion twist constant $\beta$. The transformation lies in the Ehlers orbit; our contribution is the NUT-dependent geometry and the roles of $C$ and $\beta$ in its axis, horizon, singularity, and azimuthal-CTC structure. The inversion preserves the canonical Weyl radius, the signed WLP numerator $F$, and the sign of $g_{\varphi \varphi }$ on regular domains. At the selected pole $x=\sigma$, $C=-\sigma$, the local axis condition and conicity are controlled by ${\chi }_{\sigma }^{(\beta )}=\beta -2m(2\sigma a+3l)$. Exterior zeros of the chosen seed Ernst representative obey the exact criterion $-\beta \in {\mathcal{R}}_C$; for the sampled families numerical traces yield half-line ranges with closed-form corner endpoints. For $D=a^2+l^2-2alC\mathrm{\ne }0$, ${\mathrm{\Lambda }}_{\beta }\mathrm{\Sigma }$ has a finite nonzero seed-ring limit. High-precision calculations find direction-independent finite limits of both quadratic Weyl invariants along the sampled rays, without establishing $C^2$-extendibility. At $\beta =0$, exterior simple Ernst zeros are found for the sampled $C=-1,0$ cases but not for $C=+1$, consistently with the computed ranges. Near one zero the Kretschmann scalar has a generic sixth-order blow-up; a numerical angular scan identifies exceptional directions of lower order. All sampled points in the regular ($g_{\varphi \varphi }>0$) exterior are Petrov type I. Candidate horizon locations remain those of Kerr-NUT, and the asymptotics are Levi-Civita type.