  We investigate properties of self-gravitating isorotating Skyrmions in the
generalized Einstein-Skyrme model with higher-derivative terms in the matter
field sector. These stationary solutions are axially symmetric, regular and
asymptotically flat. We provide a detailed account of the branch structure of
the spinning solutions in the topological sector of degree one. We show that
additional branches of solutions appear, as the angular frequency increases
above some critical value, these "cloudy" configurations can be considered as a
bound system of the spinning Skyrmions and "pion" excitations in the
topologically trivial sector. Considering the critical behavior of the
isorotating solitons in the general Einstein-Skyrme model, we point out that
there is no isospinning regular solutions in the reduced self-dual
$\mathcal{L}_6+\mathcal{L}_0$ submodel for any non zero value of the angular
frequency.
