  We study a new family of self-gravitating axially-symmetric, rotating boson
stars in the two-component Friedberg-Lee-Sirlin model minimally coupled to
Einstein gravity in four spacetime dimensions. In the flat space limit the
boson stars tend to the corresponding Q-balls. Subject to the usual
synchronization condition, the model admits spinning hairy black hole solutions
with two different types of scalar hair. We here investigate parity-even and
parity-odd boson stars and their associated hairy black holes. We explore the
domain of existence of the solutions and address some of their physical
properties. The solutions exhibit close similarity to the corresponding boson
stars and Kerr black holes with synchronised scalar hair in the $O(3)$-sigma
model coupled to Einstein gravity and to the corresponding solutions in the
Einstein-Klein-Gordon theory with a complex scalar field, where the latter are
recovered in a limit.
