A collection of integrable systems of the Toda type in continuous and discrete time, with 2x2 Lax representations

TL;DR

This paper provides a comprehensive list of Toda-like integrable lattice systems in continuous and discrete time, detailing their formulations, Lax representations, and r-matrix structures, without including proofs.

Contribution

It offers a systematic compilation of Toda-type integrable systems with explicit Lax pairs and Hamiltonian structures, expanding the catalog of known integrable models.

Findings

Complete list of Toda-like integrable systems

Explicit 2x2 Lax representations for each system

Detailed formulations in various mechanical frameworks

Abstract

A fairly complete list of Toda-like integrable lattice systems, both in the continuous and discrete time, is given. For each system the Newtonian, Lagrangian and Hamiltonian formulations are presented, as well as the 2x2 Lax representation and r-matrix structure. The material is given in the "no comment" style, in particular, all proofs are omitted.