  We investigate the structure of the Klein-Gordon-Fock equation symmetry
algebra on pseudo-Riemannian manifolds with motions in the presence of an
external electromagnetic field. We show that in the case of an invariant
electromagnetic field tensor, this algebra is a one-dimensional central
extension of the Lie algebra of the group of motions. Based on the coadjoint
orbit method and harmonic analysis on Lie groups, we propose a method for
integrating the Klein-Gordon-Fock equation in an external field on manifolds
with simply transitive group actions. We consider a nontrivial example on the
four-dimensional group $E(2) \times \mathbb{R}$ in detail.
