New completely integrable Neumann systems related to the perturbation KdV hierarchy

Using the nonlinearization approach, a class of new finite-dimensional Hamiltonian systems and a class of new finite-dimensional Neumann systems are obtained from the 4×4 matrix eigenvalue problem associated with the perturbation KdV hierarchy. The generating functions of integrals of motion and their generators are presented, based on which two classes of finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, involutive solutions of the perturbation KdV hierarchy are given.