  Sufficiently small Schwarzschild black holes in global AdS$_5\times$S$^5$ are
Gregory-Laflamme unstable. We construct new families of black hole solutions
that bifurcate from the onset of this instability and break the full SO$(6)$
symmetry group of the S$^5$ down to SO$(5)$. These new "lumpy" solutions are
labelled by the harmonics $\ell$. We find evidence that the $\ell = 1$ branch
never dominates the microcanonical/canonical ensembles and connects through a
topology-changing merger to a localised black hole solution with S$^8$
topology. We argue that these S$^8$ black holes should become the dominant
phase in the microcanonical ensemble for small enough energies, and that the
transition to Schwarzschild black holes is first order. Furthermore, we find
two branches of solutions with $\ell = 2$. We expect one of these branches to
connect to a solution containing two localised black holes, while the other
branch connects to a black hole solution with horizon topology $\mathrm
S^4\times\mathrm S^4$ which we call a "black belt".
