A new loop algebra and its applications to integrable system

We construct a new loop algebra which is used to set up a linear isospectral problem. By making use of Tu scheme, a new integrable couplings of the generalized MKdV hierarchy is obtained. Then its tri-Hamiltonian structure is constructed by the quadratic-form identity. Furthermore, we point out that the integrable couplings is integrable in the sense of Liouville.