We derive a dispersionless integrable system describing a local form of a general three-dimensional Einstein-Weyl geometry with an Euclidean (positive) signature, construct its matrix extension and demonstrate that it leads to the Bogomolny equations for a non-abelian monopole on an Einstein-Weyl geometry background. The corresponding dispersionless integrable hierarchy, its matrix extension and the dressing scheme are also considered.

中文翻译：

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爱因斯坦-外尔几何背景下的无色散可积系统和 Bogomolny 方程

我们推导出一个无色散可积系统，描述具有欧几里德（正）特征的一般三维爱因斯坦-外尔几何的局部形式，构造其矩阵扩展并证明它导致爱因斯坦上非阿贝尔单极子的 Bogomolny 方程-外尔几何背景。还考虑了相应的无色散可积层次、其矩阵扩展和修整方案。