  We present a new approach to the study of vacuum spacetimes with a Killing
symmetry. The central quantity in this approach is the exterior derivative of
the Killing vector field, which is a test electromagnetic field. Considering
the algebraic structure of this quantity we get a new view of the integrability
conditions, which provides a natural way of studying the connections between
the algebraic structure of the spacetime and properties of the Killing
symmetry.
