It is noted that the Niederer transformation can be used to find the explicit relation between time-dependent linear oscillators, including the most interesting case when one of them is harmonic. A geometric interpretation of this correspondence is provided by certain subclasses of pp-waves; in particular the ones strictly related to the proper conformal transformations. This observation allows us to show that the pulses of plane gravitational wave exhibiting the maximal conformal symmetry are analytically solvable. Particularly interesting is the circularly polarized family for which some aspects (such as the classical cross section, velocity memory effect and impulsive limit) are discussed in more detail. The role of the additional integrals of motion, associated with the conformal generators, is clarified by means of Ermakov-Lewis invariants. Possible applications to the description of interaction of electromagnetic beams with matter are also indicated.

We discuss in some detail the interaction of classical particles, including the scattering and memory effect, with a pulse of gravitational plane wave. The key point is the conformal symmetry of gravitational plane waves. In particular, we obtain, in the limit of short pulse, some results for impulsive gravitational waves. Furthermore, in the general case, we give certain conditions which allow us to completely describe the interaction in terms of the singular Baldwin-Jeffery-Rosen coordinates.

Using the correspondence between solutions of gravitational and gauge theories (the so-called classical double copy conjecture) some electromagnetic fields with vortices are constructed, for which the Lorentz force equations are analytically solvable. The starting point is a certain class of plane gravitational waves exhibiting the conformal symmetry. The notion of the Niederer transformation, crucial for the solvability, is analysed in the case of the Lorentz force equation on the curved spacetimes as well as its derivation by means of integrals of motion (associated with conformal generators preserving these vortices) is presented. Furthermore, some models discussed recently in the context of the intense laser beams are constructed from their gravitational counterparts, with the special emphasis put on the focusing property, and new solvable examples are presented.