  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.
