An integrable hierarchy and expanding integrable systems as well as hamiltonian structure

A Lie algebra sl(3) is first presented for which an isospectral Lax pair is introduced. The compatibility condition of the Lax pair gives rise to a new integrable hierarchy with three potential functions, call it a heat hierarchy. As its reduced cases, we present two integrable systems. The first is a generalized heat conduction equation, the second is a generalized AKNS-type hierarchy whose three kinds of Darboux transformations are obtained, which are powerful tools for generating soliton solutions of the nonlinear evolution equations from the generalized AKNS-type hierarchy. In addition, we derive the Hamiltonian structure of the heat hierarchy. Finally, by employing an enlarged Lie algebra of the Lie algebra sl(3), two isospectral problems are introduced whose compatibility condition leads to an integrable hierarchy which is an integrable couplings of the heat hierarchy. The Hamiltonian structure of the integrable couplings is worked out by taking use of the variational identity.