A hierarchy of the Lax integrable system, its bi-Hamiltonian structure, finite-dimensional integrable system and involutive solution

In this paper an isospectral problem and the associated hierarchy of Lax integrable system are considered. Zero-curvature representation and bi-Hamiltonian structures are established for the whole hierarchy by using trace identity and Lenardʹs operator pair. Moreover the isospectral problem is nonlinearized as a finite-dimensional, completely integrable Hamiltonian system under the Bargmann constraint between the potentials and the eigenvalue functions, and then an associated Lax representation is constructed. Finally finite-dimensional Liouville integrable involutive systems are found, and the involutive solutions of the hierarchy of equations are given.