"Particle-like" singular solutions in Einstein-Maxwell theory and in algebraic dynamics

Summary: "Foundations of algebrodynamics based on the previously proposed equations of biquaternionic holomorphy are briefly presented. The Maxwell and Yang-Mills equations for free fields are satisfied identically on the solutions of the primary system, which is also related to the equations of shear-free null congruences (SFC), and through them to the Einstein-Maxwell electrovacuum system. The Kerr theorem for SFC reduces the basic system to one algebraic equation, so that with each solution of the latter, some (singular) solution of the vacuum equations may be associated. We present some exact solutions of the basic algebraic and related field equations with a compact structure of singularities of the electromagnetic field, in particular, having the form of an eight-shaped curve. A fundamental solution to the primary system is analogous to the metric and fields of the Kerr-Newman solution. In addition, in the framework of algebraic dynamics the value of the electric charge for this solution is strictly fixed and may be set equal to the elementary charge."