We use a recently found method to characterise all the invertible fourth-order difference equations linear in the extremal values based on the existence of a discrete Lagrangian. We also give some result on the integrability properties of the obtained family and we put it in relation with known classifications. Finally, we discuss the continuum limits of the integrable cases.

中文翻译：

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拉格朗日和可积性四阶差分方程

基于离散拉格朗日的存在，我们使用一种最近发现的方法来刻画所有在极值上线性的可逆四阶差分方程。我们还对获得的族的可积性给出了一些结果，并将其与已知分类相关联。最后，我们讨论了可积情形的连续极限。