We study the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables. By integrability, we mean the presence of reductions of a chain to a system of hyperbolic equations of an arbitrarily high order that are integrable in the Darboux sense. Darboux integrability admits a remarkable algebraic interpretation: the Lie—Rinehart algebras related to both characteristic directions corresponding to the reduced system of hyperbolic equations must have a finite dimension. We discuss a classification algorithm based on the properties of the characteristic algebra and present some classification results. We find new examples of integrable equations.

中文翻译：

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基于 Lie-Rinehart 代数的可积二维格的分类算法

我们研究了依赖于一个离散变量和两个连续变量的非线性晶格的可积分类问题。通过可积性，我们的意思是存在一个链归约到一个任意高阶的双曲方程组的系统，这些方程在 Darboux 意义上是可积的。Darboux 可积性承认了一个非凡的代数解释：与双曲方程简化系统对应的两个特征方向相关的 Lie-Rinehart 代数必须具有有限维数。我们讨论了一种基于特征代数性质的分类算法，并给出了一些分类结果。我们找到了可积方程的新例子。