Hamiltonian forms of the two new integrable systems and two kinds of Darboux transformations

Based on the Lie algebra gl(2)gl(2), taking a kind of corresponding loop algebra gl(2)∼, a new Lax integrable hierarchy can be obtained. Then, by means of the quadratic-form identity, the corresponding bi-Hamiltonian structure was worked out. Expanding Lie algebra gl(2)gl(2), and making use of the new zero curvature equation Zhang (2008) [9], we obtain an integrable hierarchy and its Hamiltonian structure. At last, two kinds of Darboux transformations of the equation are generated.