  Starting from a completely general standpoint, we find the most general
brane-Universe solutions for a three-brane in a five dimensional spacetime. The
brane can border regions of spacetime with or without a cosmological constant.
Making no assumptions other than the usual cosmological symmetries of the
metric, we prove that the equations of motion form an integrable system, and
find the exact solution. The cosmology is indeed a boundary of a (class II)
Schwarzschild-AdS spacetime, or a Minkowski (class I) spacetime. We analyse the
various cosmological trajectories focusing particularly on those bordering
vacuum spacetimes. We find, not surprisingly, that not all cosmologies are
compatible with an asymptotically flat spacetime branch. We comment on the role
of the radion in this picture.
