We study K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-Kähler metric. One motivation for this paper is the role of such $\mathbb{Z}^2_2$-actions for the construction of Riemannian manifolds with holonomy $G_2$. We find a large class of smooth K3 surfaces with such pairs of involutions. After that, we turn our attention to the case that the K3 surface has ADE‑singularities. We introduce a special class of non-symplectic involutions that are suitable for explicit calculations and find 320 examples of pairs of involutions that act on K3 surfaces with a great variety of singularities.

中文翻译：

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具有一对交换非辛对合的 K3 曲面

我们研究具有一对通勤对合的 K3 曲面，这些对合相对于由超凯勒度量确定的两个反通勤复形结构而言是非辛的。本文的动机之一是这种 $\mathbb{Z}^2_2$ 动作在构造具有完整 $G_2$ 的黎曼流形中的作用。我们发现一大类具有此类对合的光滑 K3 曲面。之后，我们将注意力转向 K3 表面具有 ADE 奇点的情况。我们引入了一类特殊的非辛对合，它们适用于显式计算，并找到了 320 个作用于具有多种奇点的 K3 曲面上的对合示例。