Two types of hierarchies of evolution equations associated with the extended Kaup–Newell spectral problem with an arbitrary smooth function

In this paper we consider an extended Kaup–Newell (EKN) isospectral problem with an arbitrary smooth function and the corresponding two kinds of Lax integrable hierarchies by introducing two types of auxiliary spectral problems. The Hamiltonian structure of the second hierarchy is established. It is shown that the Hamiltonian system are integrable in Liouville’s sense and the set of Hamiltonian functions is the conserved densities of the second hierarchy, as well as they are in involutive in pairs under the Poisson bracket.