An integrable system and associated integrable models as well as Hamiltonian structures

Starting from an existed Lie algebra introduces a new Lie algebra A1 = {e ₁, e ₂, e ₃} so that two isospectral Lax matrices are established. By employing the Tu scheme an integrable equation hierarchy denoted by IEH is obtained from which a few reduced evolution equations are presented. One of them is the mKdV equation. The elliptic variable solutions and three kinds of Darboux transformations for one coupled equation which is from the IEH are worked out, respectively. Finally, we take use of the Lie algebra A1 to generate eight higher-dimensional Lie algebras from which the linear integrable couplings, the nonlinear integrable couplings, and the bi-integrable couplings of the IEH are engendered, whose Hamiltonian structures are also obtained by the variational identity. Then further reduce one coupled integrable equation to get a nonlinear generalized mKdV equation.