We study new families of stationary rotating soliton stars in a complex scalar field theory minimally coupled to Einstein gravity and spinning hairy black holes in the Kerr spacetime. As further examples we consider isospinning solutions of the Einstein-Skyrme model in the Kerr spacetime and similar self-gravitating solutions of 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 black holes with synchronized hairs we consider families of solutions of the Einstein-Maxwell-scalar model. Finally, we discuss Dirac stars and compare their properties to those of the usual boson stars.
