Algebraic Complete Integrability of the $a_4^{(2)}$ Toda Lattice

TL;DR

This paper proves that the $a_4^{(2)}$ Toda lattice is algebraically completely integrable by showing its generic fibers are affine parts of abelian surfaces with linear flows, and provides a geometric description of these surfaces.

Contribution

It establishes algebraic complete integrability for the $a_4^{(2)}$ Toda lattice and describes the associated abelian surfaces and their geometric properties.

Findings

Generic fibers are affine parts of abelian surfaces.

Flows of integrable vector fields are linear.

Provides geometric description of abelian surfaces and divisors.

Abstract

The aim of this work is focused on the investigation of the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra $a_4^{(2)}$. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.