Applications of an anti-symmetry loop algebra and its expanding forms

Constructing an anti-symmetry subalgebra A-tilde2 of loop algebra A-tilde2 gives the well-known Jaulent-Miodek (JM) hierarchy, the JM equation and its new Lax pair. Further, the Darboux transformation of the JM equation is deduced by anstaz method. By making use of a high-order loop algebra and Tu scheme, an expanding integrable model of the JM hierarchy is obtained. A direct expansion A-macron2* of loop algebra A-tilde2 by considering the definition of Lie algebra is presented, which is used to establish two isospectral problems. It follows that corresponding two new integrable systems are engendered, which possess bi-Hamiltonian structures, respectively. Furthermore, a scalar transformation is applied to turn the loop algebra A-bar2* into its equivalent subalgebra A-tilde1 of loop algebra A-tilde1. With the help of A-tilde1, another new high-order loop algebra G-bar is constructed, which is used to obtain an expanding integrable model of one of two integrable systems presented