Integrable Hamiltonian systems related to the polynomial eigenvalue problem

The independent integrals of motion in involution for the Hamiltonian system related to the second-order polynomial eigenvalue problem are constructed by using relevant recursion formula. The hierarchy of Hamiltonian systems obtained from the above problem and the time part of the Lax pair are shown to be completely integrable and they are shown to commute with each other. Furthermore, their solution solves the evolution equation associated with the Lax pair.