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

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.