Billiard Reprezentation for Pseudo-Euclidean Toda-Like Systems of Cosmological Origin

The pseudo-Euclidean Toda-like system of cosmological origin is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the "singularity" is reduced to a billiard on the $(n-1)$-dimensional Lobachevsky space $H^{n-1}$. The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of $(n-2)$-dimensional sphere $S^{n-2}$ by point-like sources. Some examples are considered.