  Recently it was found that the complete integration of the
Einstein-dilaton-antisymmetric form equations depending on one variable and
describing static singly charged $p$-branes leads to two and only two classes
of solutions: the standard asymptotically flat black $p$-brane and the
asymptotically non-flat $p$-brane approaching the linear dilaton background at
spatial infinity. Here we analyze this issue in more details and generalize the
corresponding uniqueness argument to the case of partially delocalized branes.
We also consider the special case of codimension one and find, in addition to
the standard domain wall, the black wall solution. Explicit relations between
our solutions and some recently found $p$-brane solutions ``with extra
parameters'' are presented.
