The aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin–Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations.

中文翻译：

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通过等距扩展的双汉密尔顿对的分类

本文的目的是对 Dubrovin-Novikov 类型的一阶哈密顿算子对进行分类，使得其中一个具有由其领先系数的等距定义的非局部部分。最近在恒定散光方程中发现了这种双汉密尔顿对的例子。我们在两个因变量的情况下获得了一个分类，以及一个具有三个因变量的重要新示例，该示例是从 Witten-Dijkgraaf-Verlinde-Verlinde 方程的特定解获得的流体动力学类型系统的扩展。