We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well-known triviality result exists, the Miura equivalence classes are parametrised by an infinite number of constants, which we call numerical invariants of the brackets. We obtain explicit formulas for the first few numerical invariants.

中文翻译：

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具有两个自变量的色散标量泊松括号的正规形式

我们用一个因变量和两个自变量对分散泊松括号进行分类，前导顺序为流体动力学类型，直到 Miura 变换。我们表明，与存在众所周知的平凡结果的单个自变量的情况相比，Miura 等价类由无限数量的常数参数化，我们称之为括号的数值不变量。我们获得了前几个数值不变量的显式公式。