Full Symmetric Toda System: Solution via QR-Decomposition

The full symmetric Toda system is a generalization of the open Toda chain, for which the Lax operator is a symmetric matrix of general form. This system is Liouville integrable and even superintegrable. Deift, Lee, Nando, and Tomei (DLNT) proposed the chopping method for constructing integrals of such a system. In the paper, a solution of Hamiltonian equations for the entire family of DLNT integrals is constructed by using the generalized QR factorization method. For this purpose, certain tensor operations on the space of Lax operators and special differential operators on the Lie algebra are introduced. Both tools can be interpreted in terms of the representation theory of the Lie algebra \(\mathfrak{sl}_n\) and are expected to generalize to arbitrary real semisimple Lie algebras. As is known, the full Toda system can be interpreted in terms of a compact Lie group and a flag space. Hopefully, the results on the trajectories of this system obtained in the paper will be useful in studying the geometry of flag spaces.