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In this report, the regularity conditions at the center for staticspherically symmetric (SSS) solutions of the Einstein equations coupled tononlinear electrodynamics (NLE) with Lagrangian $\mathcal{L}=\mathcal{L}(\mathcal{F})$, depending on the electromagnetic invariant$\mathcal{F}=F_{\mu\nu}\,F^{\mu\nu}/4$, are established. The traceless Ricci(TR) tensor eigenvalue $S$, the Weyl tensor eigenvalue $\Psi_2$ and the scalarcurvature $R$ characterize the independent Riemman tensor invariants of SSSmetrics. The necessary and sufficient regularity conditions for electric NLESSS solutions require $\lim_{r\rightarrow 0}\{\Psi_2,S,R\}\rightarrow\{0,0,(0,4\Lambda+ 4\mathcal{L}(0))\}$, such that the metric function $Q(r)$and the electric field $q_0F_{rt}=:\mathcal{E}$ behave as $\{Q,\dot Q,\ddotQ\}\rightarrow\{0,0,2\}$ and$\{\mathcal{E},\dot\mathcal{E},\ddot\mathcal{E}\}\rightarrow\{0,0,0\}$, as$r\rightarrow 0$. The general linear integral representation of the electricNLE SSS metric in terms of an arbitrary electric field $\mathcal{E}$, togetherwith $\{\Psi_2,S,R\}$, is explicitly given. Moreover, beside the regular orsingular behavior at the center, these solutions may exhibit differentasymptotic behavior at spatial infinity such as the Reissner--Nordtr\"om(Maxwell) asymptotic, or present the dS--AdS or other kind of asymptotic.

Regularity conditions for spherically symmetric solutions of Einstein-nonlinear electrodynamics equations; revised and improved version