Positive and negative nonlinear integrable lattice hierarchies are established starting from a discrete matrix spectral problem with three potential functions, particularly the corresponding Hamiltonian structures are presented respectively with the help of the trace identity, all these facts show that these two hierarchies are integrable in Liouville sense. By using Lax pair we derive infinite number of conservation laws and $N-$fold Darboux transformation (DT) for the first nontrivial system in the two hierarchies. Comparing with the usual $1-$fold DT, this kind of $N-$fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications, we derive $N-$fold explicit exact solutions from seed solutions and plot their figures with properly parameters to analyze and illustrate the propagation of solitary waves.

从具有三个势函数的离散矩阵谱问题出发，建立了正负非线性可积晶格层次，特别是借助迹线身份分别给出了相应的哈密顿结构，所有这些事实表明，这两个层次在Liouville意义上是可积的。通过使用Lax对，我们可以推导出无数的守恒定律，$ñ-$对两个层次结构中的第一个非平凡系统进行折叠Darboux变换（DT）。与平常相比$\mathrm{1个}-$折叠DT，这种 $ñ-$fold DT使我们能够生成多孤子解决方案，而无需复杂的递归过程。作为应用，我们得出$ñ-$从种子解中折叠出明确的精确解，并用适当的参数绘制其图形，以分析和说明孤立波的传播。