  For a plane gravitational wave whose profile is given, in Brinkmann
coordinates, by a $2\times2$ symmetric traceless matrix $K(U)$, the matrix
Sturm-Liouville equation $\ddot{P}=KP$ plays a multiple and central r\^ole: (i)
it determines the isometries, (ii) it appears as the key tool for switching
from Brinkmann to BJR coordinates and vice versa, (iii) it determines the
trajectories of particles initially at rest. All trajectories can be obtained
from trivial "Carrollian" ones by a suitable action of the (broken) Carrollian
isometry group.
