Infinite series solutions of the symmetry equation for the $1 +2$ dimensional continuous Toda chain

TL;DR

This paper presents explicit infinite series solutions to the symmetry equation of the 1+2 dimensional continuous Toda chain, suggesting its complete integrability.

Contribution

It provides explicit solution sequences for the symmetry equation, supporting the hypothesis of the Toda chain's complete integrability.

Findings

Explicit solutions to the symmetry equation are constructed.

The results imply the Toda chain's complete integrability.

The approach may be applicable to similar integrable systems.

Abstract

A sequence of solutions to the equation of symmetry for the continuous Toda chain in $1+2$ dimensions is represented in explicit form. This fact leads to the supposition that this equation is completely integrable.