  Existence of maximal hypersurfaces and of foliations by maximal hypersurfaces
is proven in two classes of asymptotically flat spacetimes which possess a one
parameter group of isometries whose orbits are timelike ``near infinity''. The
first class consists of strongly causal asymptotically flat spacetimes which
contain no ``black hole or white hole" (but may contain ``ergoregions" where
the Killing orbits fail to be timelike). The second class of spacetimes possess
a black hole and a white hole, with the black and white hole horizons
intersecting in a compact 2-surface $S$.
