New exact solutions of the Einstein-Maxwell equations for magnetostatic fields

The symmetry reduction method based on the Fr′echet derivative of differential operators is applied to investigate symmetries of the Einstein–Maxwell field equations for magnetostatic fields, which is a coupled system of nonlinear partial differential equations of the second order. The technique yields invariant transformations that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied to obtain the exact solutions.