  We use the classical double copy to identify a necessary condition for
Maxwell theory sources to constitute single copies of Kerr-Schild solutions to
Einstein's equations. In the case of four-dimensional Kerr-Schild spacetimes on
Minkowski backgrounds, we extend this condition to a parameterization of the
corresponding single copies. These are given by Li\'enard-Wiechert fields of
charges on complex worldlines. This unifies the known instances of the
Kerr-Schild double copy black holes on flat four-dimensional backgrounds into a
single framework. Furthermore, we use the more generic condition identified to
show why the black ring in five dimensions does not admit Kerr-Schild
coordinates.
