  In this paper we discuss some geometrical and topological properties of the
full symmetric Toda system. We show by a direct inspection that the phase
transition diagram for the full symmetric Toda system in dimensions $n=3,4$
coincides with the Hasse diagram of the Bruhat order of symmetric groups $S_3$
and $S_4$. The method we use is based on the existence of a vast collection of
invariant subvarieties of the Toda flow in orthogonal groups. We show how one
can extend it to the case of general $n$. The resulting theorem identifies the
set of singular points of $\mathrm{dim}=n$ Toda flow with the elements of the
permutation group $S_n$, so that points will be connected by a trajectory, if
and only if the corresponding elements are Bruhat comparable. We also show that
the dimension of the submanifolds, spanned by the trajectories connecting two
singular points, is equal to the length of the corresponding segment in the
Hasse diagramm. This is equivalent to the fact, that the full symmetric Toda
system is in fact a Morse-Smale system.
