The Orbit Method in the Finite Zone Integration Theory

Abstract

A construction of integrable hamiltonian systems associated with different graded realizations of untwisted loop algebras is proposed. Such systems have the form of Euler - Arnold equations on orbits of loop algebras. The proof of completeness of the integrals of motion is carried out independently of the realization of the loop algebra. The hamiltonian systems obtained are shown to coincide with hierarchies of higher stationary equations for some nonlinear PDE's integrable by inverse scattering method. We apply the general scheme for the principal and homogeneous realizations of the loop algebra $ sl_3(\R)\otimes{\cal P}(λ,λ^{-1}) $. The corresponding equations on the degenerated orbit are interpreted as the Boussinesq's and two-component modified KDV equations respectively. The scalar Lax representation for the Boussinesq's equation is found in terms of coordinates on the orbit applying the Drinfeld - Sokolov reduction procedure.