On relativistic symmetry of Finsler spaces with mutually opposite preferred directions

It is shown that in Minkowski space there exist transformations of the coordinates of events alternative to the 3-parameter Lorentz boosts. However, in contrast to the boosts, they constitute a 3-parameter noncompact group which, in turn, is a subgroup of the homogeneous 6-parameter Lorentz group. Moreover, in the same space, there exists another 3-parameter noncompact group isomorphic to above-mentioned one. As we shall see, these two 3-parameter noncompact groups are rudiments of the 3-parameter groups of relativistic symmetry of the axially symmetric Finslerian spaces with the preferred directions $\boldsymbol\nu$ and $-\boldsymbol\nu$, respectively. Finally, it will be also demonstrated that inversion of the preferred direction $\boldsymbol\nu$ in the axially symmetric Finslerian space-time does not change the Lobachevski geometry of 3-velocity space. However, this leads to an inversion of the corresponding family of horospheres of the space.