A few integrable systems and spatial spectral transformations

By using a loop algebra we obtain two new integrable hierarchies of evolution equations under the frame of zero curvature equations. The Hamiltonian structure of one of them is derived from the trace identity. By enlarging the loop algebra into two various bigger ones, two kinds of expanding integrable models of the above hierarchy with Hamiltonian structure are worked out, respectively. One has the quasi-Hamiltonian structure obtained by the variational identity, another possesses the Hamiltonian structure obtained by the quadratic-form identity. The exact computing formula for the constant γ in the variational identity is worked out smartly for given Lie algebra H. Finally, we obtain three kinds of transformations for spatial spectral problems which might be used to deduce soliton solutions.