We study new families of stationary rotating axially symmetric soliton
stars in a complex scalar field theory minimally coupled to Einstein
gravity, and spinning hairy black holes with event horizon, which
represent solutions of the Einstein-Skyrme model in the Kerr spacetime.
We found that the spinning axially symmetric cloudy solutions of the
model also exist in the regular asymptotically flat space-time. The
latter congurations are similar to the usual rotating boson stars,
which, in the flat space limit are linked to the axially symmetric
Q-balls. We found that in both cases the solutions exist for some
restricted range of values of the angular frequency and possess a
quantized angular momentum. Considering similar solution in the O(3)
sigma model, we show that, depending on the values of the parameters of
the model and the Hawking temperature, the branch structure of the
corresponding cloudy solutions varies from the usual inspirraling
pattern, which is typical for the boson stars, to the two branch
structure, similar to that of the black holes with Skyrme hairs. As
another examples of spinning black holes with synchronized hairs we
consider families of parity-odd solutions of the Einstein-Klein-Gordon
model. Finally, we discuss Dirac stars and compare their properties with
the usual boson stars.