We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes of polynomial matrices. Let X be the algebraic curve given by the common characteristic equation for . 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 an application, we discuss the algebraic complete integrability of the extended Lotka–Volterra lattice with a periodic boundary condition.