We first introduce a Lie algebra $\tilde{g}$ which can be used to construct integrable couplings of some isospectral and nonisospectral problems. As two applications of the Lie algebra $\tilde{g},$ the MKdV spectral problem is enlarged to an isospectral problem and the AKNS spectral problem is expanded to a nonisopectral problem. Then, two integrable couplings are obtained by solving an isospectral and a nonisospectral zero-curvature equations. We find that the two hierarchies that we obtain have bi-Hamiltonian structure of combinatorial form. Additionally, some symmetries and conserved quantities of the resulting hierarchy are investigated.

中文翻译：

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一类非等谱与等谱可积耦合及其哈密顿系统。

我们首先介绍一个李代数 $\stackrel{〜}{G}$可以用来构造一些等光谱和非等光谱问题的可积耦合。作为李代数的两个应用$\stackrel{〜}{G}，$MKdV光谱问题扩展为等光谱问题，而AKNS光谱问题扩展为非光谱问题。然后，通过求解一个等光谱和一个非等光谱的零曲率方程，获得了两个可积分耦合。我们发现，我们获得的两个层次结构具有组合形式的双哈密尔顿结构。此外，研究了所得层次结构的一些对称性和守恒量。