Geometry of the full symmetric Toda system

Full symmetric Toda system is the Lax-type system $\dot L=[M(L),L]$, where the variable $L$ is a real symmetric $n\times n$ matrix and $M(L)=L_+-L_-$ denotes its "naive" anti-symmetrisation, i.e. the matrix constructed by taking the difference of strictly upper- and lower-triangular parts $L_+$ and $L_-$ of $L$. This system has lots of interesting properties: it is a Liouville-integrable Hamiltonian system (with rational first integrals), it is also super-integrable (in the sense of Nekhoroshev), its singular points and trajectories represent the Hasse diagram of Bruhat order on permutations group. Its generalizations to other semisimple real Lie algebras have similar properties. In my talk I will sketch the proof of some of these properties and will describe a construction of infinitesimal symmetries of the Toda system. It turns out that there are many such symmetries, their construction depends on representations of $\mathfrak{sl}_n$. As a byproduct we prove that the full symmetric Toda system is integrable in the sense of Lie-Bianchi criterion.
The talk is based on a series of papers joint with Yu.Chernyakov, D.Talalaev and A.Sorin.