In this article there is described an analytical approach to describe the self-field of two- and three dimensional ellipsoidal presentation of space charge distribution. The corresponding results can be evaluated in both numerical and the analytic presentation for some model distributions of charge. The corresponding results can be embedded in the Lie formalism used to describe the map for the beam dynamics. The corresponding linear and nonlinear maps are evaluated in terms of the matrix representation of the evolution operator of the beam. Appropriate solutions for nonlinear differential equations are based on a prediction-correction method (the converging recursive procedure). These solutions are compared with the Vlasov equation solutions. A special software package for the described approach is presented.