Exact solutions of the Klein-Gordon equation in external electromagnetic fields on 3D de Sitter background

In this study, we investigate the symmetry properties and the possibility of exact integration of the Klein--Gordon equation in the presence of an external electromagnetic field on 3D de Sitter background. 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 inequivalent subalgebras of the algebra $\mathfrak{so}(1,3)$, we obtain the classification of the electromagnetic fields on $\mathrm{dS}_3$ admitting first-order symmetry algebras of the Klein-Gordon equation. Then, we select the integrable cases, and for each of them, we construct exact solutions, using the non-commutative integration method developed by Shapovalov and Shirokov. In Appendix, we present an original algebraic method for constructing the special local coordinates on de Sitter space, in which the basis vector fields for subalgebras of the algebra $\mathfrak{so}(1,3)$ have the simplest form.