A generalized Broer–Kaup (GBK) hierarchy and its expanding integrable system

A new Lax pair is first constructed. By making use of Tu scheme, a Lax integrable system is engendered. Since it can reduce to a generalized Broer–Kaup (GBK) system, we call it GBK hierarchy. Second, both Darboux transformations of the GBK system are obtained, which can generate new solutions. At last, an expanding integrable system of the GBK hierarchy, which is also an integrable coupling, is presented by using the direct sum relations and isomorphic relations between two subalgebras of a high order loop algebra G∼.