The Lie -Bianchi integrability of the full symmetric Toda system

TL;DR

This paper proves the integrability of the full symmetric Toda system using Lie-Bianchi criteria, revealing a solvable Lie algebra of symmetries that leave the system invariant.

Contribution

It establishes the Lie-Bianchi integrability of the full symmetric Toda system and identifies the structure of the associated stochastic Lie algebra.

Findings

Full symmetric Toda system is Lie-Bianchi integrable.

Existence of a solvable Lie algebra of symmetries for the system.

Connection between symmetries and stochastic Lie algebra structure.

Abstract

In this paper we prove that the full symmetric Toda system is integrable in the sense of the Lie-Bianchi criterion, i.e. that there exists a solvable Lie algebra of vector fields of dimension $N=\dim M$ on the phase space $M$ of this system such that the system is invariant with respect to the action of these fields. The proof is based on the use of symmetries of the full symmetric system, which we described earlier in \cite{CSS23}, and the appearance of the structure of the stochastic Lie algebra in their description.