A finite-dimensional integrable system associated with a polynomial eigenvalue problem

M. Antonowicz and A. P. Fordy (1988) introduced the second-order polynomial eigenvalue problem Lφ = (∂2 +∑i=1 viλ)φ = αφ (∂ = ∂/∂x, α = constant) and discussed its multi-Hamiltonian structures. For n= 1 and n= 2, the associated finite-dimensional integrable Hamiltonian systems (FDIHS) have been discussed by Xu and Mu (1990) using the nonlinearization method and Bargmann constraints. In this paper, we consider the general case, that is, n is arbitrary, provide the constrained Hamiltonian systems associated with the above-mentioned second-order polynomial ergenvalue problem, and prove them to be completely integrable.