Abstract. The aim of this paper is to generate a kind of integrable hierarchies of four-component evolution equations with Hamiltonian structures, from a kind of reduced Ablowitz-Kaup-Newell-Segur (AKNS) matrix spectral problems. The zero curvature formulation is the basic tool and the trace identity is the key to establishing Hamiltonian structures. Two examples of Hamiltonian equations in the resulting integrable hierarchies are added to the category of coupled integrable nonlinear Schrödinger equations and coupled integable modified Korteweg-de Vries equations.

Key words: Integrable hierarchy, Lax pair, Zero curvature equation, NLS equations, mKdV equations.