The C.Neumann System And Integrable System Belated To A Nonself-Adjoint Third-Order Eigenvalue Problem

The evolution equations and the integrablity of the third-order eigenvalue problem L(?) = (-(?)3 + q(?) + (?)q +p)(?) = λ(?) are open problems.In this paper, By means of the C.Neumann constraint Γ and the relations between the potentials (q,p) and the eigenvectors, based on the viewpoint of Hamiltonian mechanics and the Euler-Lagrange equations and Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system has been found. The Lax pairs of the good-Boussinesq equation hierarchy are nonlinearized, and according to the Moser restrict, a finite-dimensional integrable system in the Liouville sense is generated. By the relations between the potentials (q,p) and the eigenvectors, the involutive representations of the solutions for the good-Boussinesq equation hierarchy are obtained.