Scaling and conformal symmetries for plane gravitational waves

Isometries of an exact plane gravitational wave spacetime are symmetries for both massive and massless particles. Their conformal extensions, among which Chrono-Projective transformations play a distinguished role, are symmetries for massless particles. Homotheties are universal symmetries for any profile. The generically 5 parameter isometry group is extended Chrono-Projectively to a 6 or 7 parameter symmetry group under special circumstances.