  We consider the vacuum Einstein field equations under the Belinski-Zakharov
symmetries. Depending on the chosen signature of the metric, these spacetimes
contain most of the well-known special solutions in General Relativity,
including well-known black holes. In this paper, we prove global existence of
small Belinski-Zakharov spacetimes under a natural nondegeneracy condition. We
also construct new energies and virial functionals to provide a description of
the energy decay of smooth global cosmological metrics inside the light cone.
Finally, some applications are presented in the case of generalized Kasner
solitons.
