On Killing vectors and invariance transformations of the Einstein-Maxwell equations

It is shown that, in a simply connected four dimensional Riemannian space, an arbitrary divergence-free vector generates a one-parameter group of point transformations which leaves Maxwell's equations unchanged. This result is used to show that, if the metric tensor of a simply connected vacuum Einstein-Maxwell space-time admits a group of motions which is also an invariance group of the electro-magnetic field tensor, then there exists a one-parameter family of metric tensors all of which satisfy the Einstein-Maxwell equations with the invariant electromagnetic field as source. (author)