To study the reduced fourth-order eigenvalue problem, the Bargmann constraint of this problem has been given, and the associated Lax pairs have been nonlineared. By means of the viewpoint of Hamilton mechanics, the Euler-Lagrange function and the Legendre transformations have been derived, and a reasonable Jacobi-Ostrogradsky coordinate system has been found. Then, the Hamiltonian cannonical coordinate system equivalent to this eigenvalue problem has been obtained on the symplectic manifolds. It is proved to be an infinite-dimensional integrable Hamilton system in the Liouville sense. Moreover the involutive representation of the solutions is generated for the evolution equations hierarchy in correspondence with this reduced fourth-order eigenvalue problem.