  We enlarge the spectral problem of a generalized D-Kaup-Newell (D-KN)
spectral problem. Solving the enlarged zero-curvature equations, we produce
integrable couplings. A reduction of the spectral matrix leads to a second
integrable coupling system. Next, bilinear forms that are symmetric,
ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra
are computed to employ the variational identity. The variational identity is
then applied to the original enlarged spectral problem of a generalized D-KN
hierarchy and the reduced problem. Hamiltonian structures are presented, as
well as a bi-Hamiltonian formulation of the reduced problem. Both hierarchies
have infinitely many commuting symmetries and conserved densities, i.e., are
Liouville integrable.
