  In a space-time $M$ with a Killing vector field $\xi^a$ which is either
everywhere timelike or everywhere spacelike, the collection of all trajectories
of $\xi^a$ gives a 3-dimension space $S$. Besides the symmetry-reduced action
from that of Einstein-Hilbert, an alternative action of the fields on $S$ is
also proposed, which gives the same fields equations as those reduced from the
vacuum Einstein equation on $M$.
