The matrix realization of affine Jacobi varieties and the extended Lotka-Volterra lattice

We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve given by the common characteristic equation for $\boldsymbol{\mathcal{M}}_F$. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of $X$. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As the application, we discuss the algebraic completely integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.