Generalization of perturbation theory for a nonlinear system, losing a small parameter in the subregion where the solution changes

The paper examines a nonlinear system of equations of motion for a material point (or charge) in a force field under the action of a continuous, initially small, perturbing force, whose relation to the gravitational (or attractive) force assumes such a form, that the system loses a small parameter in the course of time. The perturbation does not depend on fast time, but only on coordinates, velocities, and slow time. A method is proposed for obtaining an approximate solution unique for all regions where the small parameter is present and where it vanishes from the system.