  The isometries of an exact plane gravitational wave are symmetries for both
massive and massless particles. Their conformal extensions are in fact
chrono-projective transformations introduced earlier by Duval et al are
symmetries for massless particles. Homotheties are universal chrono-projective
symmetries for any profile. Chrono-projective transformations also generate new
conserved quantities for the underlying non-relativistic systems in the
Bargmann framework. Homotheties play a similar role for the lightlike
"vertical" coordinate as isometries play for the transverse coordinates.
