A new non-semisimple Lie algebra $\widehat{g}$ is constructed. Based on the Lie algebra $\widehat{g}$, we propose a method for generating $Z_N^{\varepsilon }$ integrable couplings. Then, we consider the application of a MKdV isospectral and an AKNS nonisospectral problem of the method respectively. By solving the zero curvature equation of the two spectral problems, we obtain a $Z_N^{\varepsilon }$ isospectral MKdV integrable coupling hierarchy and a $Z_N^{\varepsilon }$ nonisospectral ANKS integrable coupling hierarchy. It follows that the bi-Hamiltonian structures of these two hierarchies is derived based on the $Z_N^{\varepsilon }$-trace identity that we constructed.

中文翻译：

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一种新的多分量可积耦合及其在等谱和非等谱问题中的应用

一个新的非半简单李代数 $\widehat{G}$被构造。基于李代数$\widehat{G}$，我们提出了一种生成方法 $Z_N^{\varepsilon }$可集成联轴器。然后，我们分别考虑该方法的 MKdV 等谱问题和 AKNS 非等谱问题的应用。通过求解两个谱问题的零曲率方程，我们得到一个$Z_N^{\varepsilon }$ 等谱 MKdV 可积耦合层次和 $Z_N^{\varepsilon }$非等谱 ANKS 可积耦合层次。因此，这两个层次结构的双汉密尔顿结构是基于$Z_N^{\varepsilon }$-我们构建的trace身份。