标题： 线性化、可分离性和$a_4^{(2)}$陶布斯格子的Lax对表示 显示英文标题

标题： Linearization, separability and Lax pairs representation of $a_4^{(2)}$ Toda lattice

摘要： 本工作的目的是对与扭曲仿射李代数\(a_4^{\left(2\right)}\)相关的代数完全可积(Toda格子)进行线性化并找到其Lax对。 首先，我们回顾到我们的a.c.i情况是一个二维代数完全可积系统，其中不变(实)环面可以扩展为复代数环面(阿贝尔曲面)。 这意味着可以利用几何方法来研究该系统。 其次，我们表明该格子与Mumford系统相关，并在这两个系统之间构造了一个显式的同态，从而为Mumford系统带来了新的泊松结构。 最后，我们给出了这个Toda格子的新Lax方程，并构造了该系统的显式线性化。 显示英文摘要

摘要： The aim of this work is focused on linearizing and found the Lax Pairs of the algebraic complete integrability (a.c.i) Toda lattice associated with the twisted affine Lie algebra \(a_4^{\left(2\right)}\). Firstly, we recall that our case of a.c.i is a two-dimensional algebraic completely integrable systems for which the invariant (real) tori can be extended to complex algebraic tori (abelian surfaces). This implies that the geometry can be used to study this system. Secondly, we show that the lattice is related to the Mumford system and we construct an explicit morphism between these systems, leading to a new Poisson structure for the Mumford system. Finally, we give a new Lax equation for this Toda lattice and we construct an explicit linearization of the system.