Conformally related vacuum gravitational waves, and their symmetries

A special conformal transformation which carries a vacuum gravitational wave into another vacuum one is built by using the Möbius-redefined time. It can either interchange two different vacuum waves transforming a global one into a sandwich vacuum wave, or transform the gravitational wave into itself. The first type, illustrated by linearly and circularly polarized vacuum plane gravitational waves, interchanges the symmetries and the geodesics. The second type, inspired by molecular physics, is illustrated by an anisotropic generalization of the familiar inverse-square profile with its familiar conformal O(2,1) symmetry. The particle can escape, or perform periodic motion, or fall into the singularity.