  We study solutions of the Einstein-Skyrme model. Firstly we consider test
field Skyrmions on the Kerr background. These configurations -- hereafter
dubbed Skerrmions -- can be in equilibrium with a Kerr black hole (BH) by
virtue of a synchronisation condition. We consider two sectors for Skerrmions.
In the sector with non-zero baryon charge, Skerrmions are akin to the known
Skyrme solutions on the Schwarzschild background. These `topological'
configurations reduce to flat spacetime Skyrmions in a vanishing BH mass limit;
moreoever, they never become "small" perturbations on the Kerr background: the
non-linearities of the Skyrme model are crucial for all such Skerrmions. In the
non-topological sector, on the other hand, Skerrmions have no analogue on the
Schwarzschild background. Non-topological Skerrmions carry not baryon charge
and bifurcate from a subset of Kerr solutions defining an existence line.
Therein the appropriate truncation of the Skyrme model yield a linear scalar
field theory containing a complex plus a real field, both massive and
decoupled, and the Skerrmions reduce to the known stationary scalar clouds
around Kerr BHs. Moreover, non-topological Skerrmions trivialise in the
vanishing BH mass limit. We then discuss the backreaction of these Skerrmions,
that yield rotating BHs with synchronised Skyrme hair, which continously
connect to the Kerr solution (self-gravitating Skyrmions) in the
non-topological (topological) sector. In particular, the non-topological hairy
BHs provide a non-linear realisation, within the Skyrme model, of the
synchronous stationary scalar clouds around Kerr.
