  We examine the Hamiltonian formulation and hyperbolicity of the
almost-Killing equation (AKE). We find that for all but one parameter choice,
the Hamiltonian is unbounded, and in some cases, the AKE has ghost degrees of
freedom. We also show the AKE is only strongly hyperbolic for one parameter
choice, which corresponds to a case in which the AKE has ghosts. Fortunately,
one finds that the AKE reduces to the homogeneous Maxwell equation in a vacuum,
so that with the addition of the divergence-free constraint (a "Lorenz gauge"),
one can still obtain a well-posed problem that is stable in the sense that the
corresponding Hamiltonian is positive definite. An analysis of the resulting
Komar currents reveals an exact Gauss law for a system of black holes in vacuum
spacetimes and suggests a possible measure of matter content in asymptotically
flat spacetimes.
