A matrix integrable Hamiltonian hierarchy and its two integrable reductions

A higher-order spectral problem is introduced, based on a special Lie subalgebra of the general linear algebra. An associated matrix Liouville integrable hierarchy, each of which consists of four submatrix equations, is generated by means of the zero curvature formulation. The corresponding Hamiltonian structure is established by the trace variational identity and two integrable reductions, of which one is real and the other is complex, are constructed under similarity transformations.