New extended Lie algebra and the generalized integrable Liouville hierarchy

Starting from a new Lie algebra G1, the corresponding loop algebra G1∼ is presented, from which a Liouville integrable hierarchy is given by using of variational identity. It follows that an expanding Lie algebra G2 is obtained based on G1, furthermore, a related Lax integrable hierarchy is presented by making use of its related loop algebra G2∼. With the help of variational identity, it is not difficult to prove that the hierarchy has Hamiltonian structure and is Liouville integrable. It is also can be seen that the second hierarchy is the expanding model of the first one.