  We reinvestigate the utility of time-independent constant mean curvature
foliations for the numerical simulation of a single spherically-symmetric black
hole. Each spacelike hypersurface of such a foliation is endowed with the same
constant value of the trace of the extrinsic curvature tensor, $K$. Of the
three families of $K$-constant surfaces possible (classified according to their
asymptotic behaviors), we single out a sub-family of singularity-avoiding
surfaces that may be particularly useful, and provide an analytic expression
for the closest approach such surfaces make to the singularity. We then utilize
a non-zero shift to yield families of $K$-constant surfaces which (1) avoid the
black hole singularity, and thus the need to excise the singularity, (2) are
asymptotically null, aiding in gravity wave extraction, (3) cover the
physically relevant part of the spacetime, (4) are well behaved (regular)
across the horizon, and (5) are static under evolution, and therefore have no
``grid stretching/sucking'' pathologies. Preliminary numerical runs demonstrate
that we can stably evolve a single spherically-symmetric static black hole
using this foliation. We wish to emphasize that this coordinatization produces
$K$-constant surfaces for a single black hole spacetime that are regular,
static and stable throughout their evolution.
