Regular and singular solutions for charged dust distributions in the Einstein--Maxwell theory

Solutions for the static spherically symmetric extremally charged dust in the Majumdar--Papapetrou system have been found. For a certain amount of the allocated mass/charge, the solutions have singularities of a type which could render them physically unacceptable, since the corresponding physically relevant quantities are singular as well. These solutions, with a number of zero-nodes in the metric tensor, are regularized through the $\delta$-shell formalism, thus redefining the mass/charge distributions. The bifurcating behaviour of regular solutions found before is no longer present in these singular solutions, but quantized-like behaviour in the total mass is observed. Spectrum of regularized solutions restores the equality of the Tolman--Whittaker and ADM mass, as well the equality of the net charge and ADM mass, which is the distinctive feature of Majumdar--Papapetrou systems.