  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.
