We study the exact solutions of quantum integrable model associated with the $C_n$ Lie algebra, with either a periodic or an open one with off-diagonal boundary reflections, by generalizing the nested off-diagonal Bethe ansatz method. Taking the $C_3$ as an example we demonstrate how the generalized method works. We give the fusion structures of the model and provide a way to close fusion processes. Based on the resulted operator product identities among fused transfer matrices and some necessary additional constraints such as asymptotic behaviors and relations at some special points, we obtain the eigenvalues of transfer matrices and parameterize them as homogeneous $T-Q$ relations in the periodic case or inhomogeneous ones in the open case. We also give the exact solutions of the $C_n$ model with an off-diagonal open boundary condition. The method and results in this paper can be generalized to other high rank integrable models associated with other Lie algebras.

我们研究了与量子相关的量子可积模型的精确解 $C_{ñ}$通过泛化嵌套的非对角Bethe ansatz方法，具有周期性或开放性且具有非对角边界反射的李代数。以$C_3$作为示例，我们演示了广义方法的工作原理。我们给出了模型的融合结构，并提供了一种关闭融合过程的方法。基于融合传递矩阵之间的算符乘积恒等式以及某些必要的附加约束（例如某些特殊点处的渐近行为和关系），我们获得传递矩阵的特征值并将其参数化为齐次$Ť-问$周期性情况下的关系或开放情况下的不均匀关系。我们还给出了确切的解决方案$C_{ñ}$具有非对角线开放边界条件的模型。本文的方法和结果可以推广到与其他李代数相关的其他高阶可积模型。