  We study the free motion of a massive particle moving in the background of a
Finslerian deformation of a plane gravitational wave in Einstein's General
Relativity. The deformation is a curved version of a one-parameter family of
Relativistic Finsler structures introduced by Bogoslovsky, which are invariant
under a certain deformation of Cohen and Glashow's Very Special Relativity
group ISIM(2). The partially broken Carroll Symmetry we derive using
Baldwin-Jeffery-Rosen coordinates allows us to integrate the geodesics
equations. The transverse coordinates of timelike Finsler-geodesics are
identical to those of the underlying plane gravitational wave for any value of
the Bogoslovsky-Finsler parameter $b$. We then replace the underlying plane
gravitational wave by a homogenous pp-wave solution of the Einstein-Maxwell
equations. We conclude by extending the theory to the
Finsler-Friedmann-Lemaitre model.
