We introduce a Lie algebra \(\widetilde {g}\) which can be used to construct integrable couplings of some spectral problems. As two examples, the non-semisimple Lie algebra \(\widetilde {g}\) is applied to enlarge the spectral problems of an extended Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem and a generalized D-Kaup-Newell (D-KN) spectral problem. It follows that we obtain two generalized integrable couplings by solving these expanded zero-curvature equations. Finally, we find that the integrable hierarchies that we obtain have bi-Hamiltonian structures of combinatorial form, thereby showing their Liouville integrability.

我们介绍了一个李代数\（\ widetilde {g} \），该代数可用于构造某些频谱问题的可积耦合。作为两个示例，非半简单李代数\ {\ widetilde {g} \）用于扩大扩展的Ablowitz-Kaup-Newell-Segur（AKNS）光谱问题和广义D-Kaup-Newell（ D-KN）频谱问题。因此，通过求解这些扩展的零曲率方程，我们获得了两个广义可积耦合。最后，我们发现我们获得的可积层次具有组合形式的双哈密顿结构，从而显示出它们的Liouville可积性。