We study symmetry properties and the possibility of exact integration of Klein–Gordon equations in external electromagnetic fields on 3D de Sitter background dS₃. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra extensions. Based on the well-known classification of the subalgebras of the algebra $\mathfrak{s}\mathfrak{o}(1,3)$, we classify all electromagnetic fields on dS₃ for which the corresponding Klein–Gordon equations admit first-order symmetry algebras. Then, we select the integrable cases, and for each of them, we construct exact solutions using the noncommutative integration method developed by Shapovalov and Shirokov [Theor. Math. Phys. 104, 921–934 (1995)]. We also propose an original algebraic method for constructing the special local coordinates on de Sitter space dS₃, in which basis vector fields for subalgebras of the Lie algebra $\mathfrak{s}\mathfrak{o}(1,3)$ have the simplest form.

中文翻译：

---

3D de Sitter 背景下外部电磁场中 Klein-Gordon 方程的精确解

我们研究了对称特性以及在 3D de Sitter 背景 dS ₃上的外部电磁场中精确积分 Klein-Gordon 方程的可能性。我们提出了一种构造一阶对称代数的算法，并根据李代数扩展描述了其结构。基于众所周知的代数子代数分类$\mathfrak{秒}\mathfrak{Ø}(1,3)$，我们对 dS ₃上的所有电磁场进行分类，其对应的 Klein-Gordon 方程承认一阶对称代数。然后，我们选择可积的情况，对于它们中的每一个，我们使用 Shapovalov 和 Shirokov [Theor. 数学。物理。104 , 921–934 (1995)]。我们还提出了一种在 de Sitter 空间 dS ₃上构造特殊局部坐标的原始代数方法，其中李代数的子代数的基向量场$\mathfrak{秒}\mathfrak{Ø}(1,3)$ 有最简单的形式。