We propose two different appraoches to extending the Gambier mapping to a two-dimensional lattice equation. A first approach relies on a hypothesis of separate evolutions in each of the two directions. We show that known equations like the Startsev-Garifullin-Yamilov equation, the Hietarinta equation, as well as one proposed by the current authors, are in fact Gambier lattices. A second approach, based on the same principle as the Gambier equation, that of two linearisable equations in cascade, constructs a Gambier lattice in the form of a system of two coupled Burgers equations. The (slow) growth properties of the latter are in agreement with its linearisable character.

中文翻译：

---

甘比尔格和其他可线性化系统

我们提出了两种不同的方法来将 Gambier 映射扩展到二维格子方程。第一种方法依赖于在两个方向中的每一个方向上单独进化的假设。我们证明了已知的方程，如 Startsev-Garifullin-Yamilov 方程、Hietarinta 方程以及当前作者提出的方程，实际上是 Gambier 格。第二种方法基于与 Gambier 方程相同的原理，即两个可线性化的级联方程，以两个耦合 Burgers 方程组的形式构建 Gambier 格。后者的（缓慢）增长特性与其线性化特性一致。