  It is well known that static spherically symmetric spacetimes can admit
foliations by flat spacelike hypersurfaces, which are best described in terms
of the Painlev\`{e}--Gullstrand coordinates. The uniqueness and existence of
such foliations were addressed earlier. In this paper, we prove, purely
geometrically, that any possible foliation of a static spherically symmetric
spacetime by an arbitrary codimension-one spherical spacelike geometry, up to
time translation and rotation, is unique, and we find the algebraic condition
under which it exists. This leads us to what can be considered as the most
natural generalization of the Painlev\`{e}--Gullstrand coordinate system for
static spherically symmetric metrics, which, in turn, makes it easy to derive
generic conclusions on foliation and to study specific cases as well as to
easily reproduce previously obtained generalizations as special cases. In
particular, we note that the existence of foliation by flat hypersurfaces
guarantees the existence of foliation by hypersurfaces whose Ricci curvature
tensor is everywhere non-positive (constant negative curvature is a special
case). The study of uniqueness and the existence concurrently solves the
question of embeddability of a spherical spacelike geometry in one-dimensional
higher static spherically symmetric spacetimes, and this produces known and new
results geometrically, without having to go through the momentum and
Hamiltonian constraints.
