ИСТИНА

The talk is based on a joint paper with Yu.Chernyakov (ITEP) and A.Sorin (JINR). We consider the full symmetric (nonperiodic) Toda chain. We show, that the stable points of this system are equal to the diagonal matrices with same eigenvalues, written in different order. We also show that two such matrices are connected by a trajectory, iff the corresponding permutations are compareble in Bruhat order on the group of permutations and trajectories in this case span a submanifold, whose dimension is determined by the lengths of the corresponding permutations, so that the system has a Morse-Smale structure (if we pass to suitable coordinates).