A Lax integrable hierarchy, N-Hamiltonian structure, r-matrix, finite-dimensional Liouville integrable involutive systems, and involutive solutions

An eigenvalue problem and the associated new Lax integrable hierarchy of nonlinear evolution equations are presented in this paper. As a reduction, a representative system of the generalized derivative nonlinear Schrödinger equations in the hierarchy is obtained. Zero curvature representation and N-Hamiltonian structures are established for the whole hierarchy based upon N pairs of Hamiltonian operators (Lenardʹs operators), and it is shown that the hierarchy of nonlinear evolution equations is integrable in Liouvilleʹs sense. Thus the hierarchy of equations has infinitely many commuting symmetries and conservation laws. Moreover the eigenvalue problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system under the Bargmann constraint between the potentials and the eigenvalue functions, and then an associated nondynamical modified r-matrix is constructed. Finally finite-dimensional Liouville integrable involutive systems are found, and the involutive solutions of the hierarchy of equations are given, in particular, the involutive solutions are developed for the system of generalized derivative nonlinear Schrödinger equations.