We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability is that of “multidimensional consistency” (MDC): it should be possible to extend the equation from two to higher dimensions so that the embedded two-dimensional lattice equations are compatible. Usually compatibility is checked using “Consistency-Around-a-Cube” (CAC). In this context it is often assumed that the equations on the six sides of the cube are the same (up to lattice parameters), but this assumption was relaxed in the classification of Boll [3]. We present here the results of a search and classification of homogeneous quadratic triplets of multidimensionally consistent lattice equations, allowing different equations on the three orthogonal planes (hence triplets) but using the same equation on parallel planes. No assumptions are made about symmetry or tetrahedron property. The results are then grouped by subset/limit properties, and analyzed by the effectiveness of their Bäcklund transformations, or equivalently, by the quality of their Lax pair (fake or not).

中文翻译：

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搜索四元方程的 CAC 可积齐次二次三元组及其通过 BT 和 Lax 的分类

我们考虑在笛卡尔格子的基本正方形上定义的二维格子方程，并取决于四边形角处的变量。对于此类方程，通常与可积性相关的属性是“多维一致性”（MDC）：应该可以将方程从二维扩展到更高维，以便嵌入的二维晶格方程是兼容的。通常使用“Consistency-Around-a-Cube”（CAC）检查兼容性。在这种情况下，通常假设立方体六个边上的方程是相同的（直到晶格参数），但在 Boll [3] 的分类中放宽了这一假设。我们在这里展示了多维一致格子方程的齐次二次三元组的搜索和分类结果，允许在三个正交平面上使用不同的方程（因此是三元组），但在平行平面上使用相同的方程。没有对对称性或四面体性质做出任何假设。然后将结果按子集/极限属性分组，并通过其 Bäcklund 变换的有效性，或等效地，通过其 Lax 对（假与否）的质量进行分析。