  The paper is devoted to the algebraic and geometric aspects of the full
symmetric Toda system. We construct a solution to the complete
Deift-Li-Nanda-Tomei flows system using the QR decomposition method. For this
purpose we introduce specialized invariant tensor operations on the Lax
operator of the model. These operations have a direct interpretation in terms
of the representation theory of Lie algebras. We expect that this approach can
be effective in studying the geometry of flag varieties.
