The Hamiltonian System Associated with the Third-Order Eigenvalue Problem

By means of the nonlinearized Lax pairs,the Hamiltonian system associated with a third-order spectral problem is discussed. Based on the Arnold theorem and the generating function,the Hamiltonian system is completely integrable in the Liouville sense. Moreover,according to the constrained condition between the potential function and the eigenfunction,the involutive solutions of the evolution equations are given.