Algebraic integrability and geometry of the $\mathfrak{d}_3^{(2)}$ toda lattice

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In this paper, we consider the Toda lattice associated to the twisted affine Lie algebra \(\mathfrak{d}_3^{(2)}\). We show that the generic fiber of the momentum map of this system is an affine part of an Abelian surface and that the flows of integrable vector fields are linear on this surface, so that the system is algebraic completely integrable. We also give a detailed geometric description of these Abelian surfaces and of the divisor at infinity. As an application, 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 with spectral parameter for this Toda lattice and we construct an explicit linearization of the system.