A Liouville integrable hierarchy, symmetry constraint, new finite-dimensional integrable systems, involutive solution and expanding integrable models

A new spectral problem and the associated integrable hierarchy of nonlinear evolution equations are presented in this paper. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. An explicit symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of the hierarchy. Moreover, the corresponding Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative, new finite-dimensional completely integrable Hamiltonian systems in the Liouville sense. Further, an involutive representation of solution of each equation in the hierarchy is given. Finally, expanding integrable models of the hierarchy are constructed by using a new Loop algebra.