We consider the class of regular axially symmetric solutions to the Einstein equations specified by T_t^t = T_r^r (p_r = −ρ), which describe neutral and electrically charged rotating black holes and solitons (structures without horizons), whose interiors contain characteristic surfaces filled with de Sitter vacuum. Neutral regular structures are asymptotically Kerr for a distant observer. Open question concerns a kind of a matter behind the de Sitter vacuum surface, where the weak energy condition is violated. In nonlinear electrodynamics minimally coupled to gravity, regular axially symmetric solutions to the dynamical equations are asymptotically Kerr-Newman for a distant observer. Electromagnetic field always satisfies the condition p_r=−ρ and the existence of the interior de Sitter surfaces is obligatory. They have properties of a perfect conductor and ideal diamagnetic, the ring singularity is replaced with a superconducting current, and violation of the weak energy condition can be prevented by the basic requirements of electrodynamics of continued media. Open question concerns a case when (if) these requirements can be avoided.

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