Observing the displacement of particles by a Gravitational Wave could provide a way to detect the latter. Following Souriau (1973), the geodesic equations can be integrated by using the 5-parameter isometries of plane gravitational waves, identified more recently as Lévy-Leblond's (1965) ``Carroll'' group in 2+1 dimensions with no rotations, which acts as a symmetry for the particles. The associated conserved quantities determine the trajectories.
Joint work with C. Duval, M. Elbistan, G. W.Gibbons, P.M.Zhang