  Maxwell's vacuum equations are integrated for admissible electromagnetic
fields in homogeneous spaces. Admissible electromagnetic fields are those for
which the space group generates an algebra of symmetry operators ( integrals of
motion ) that is isomorphic to the algebra of group operators. Two frames
associated with the group of motions are used to obtain systems of ordinary
differential equations to which Maxwell's equations reduce. The solutions are
obtained in quadratures. The potentials of the admissible electromagnetic
fields and the metrics of the spaces contained in the obtained solutions depend
on six arbitrary time functions, so it is possible to use them to integrate
field equations in the theory of gravity.
