Algebras of integrals of motion for the Hamilton–Jacobi and Klein–Gordon–Fock equations in spacetime with four-parameter groups of motions in the presence of an external electromagnetic field

Abstract

The algebras of the integrals of motion of the Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged test particle moving in an external electromagnetic field in a spacetime manifold are found. The manifold admits four-parameter groups of motions that act nontransitively on the spacetime. All admissible electromagnetic fields for which such algebras exist are found. In the case of an arbitrary n-dimensional Riemannian space on which the groups of motions act, it is proved that the admissible field does not deform the algebra of symmetry operators of the free Hamilton–Jacobi and Klein–Gordon–Fock equations. In addition, the system of differential equations, which must be satisfied by the potentials of the admissible electromagnetic field, has been investigated for compatibility.