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

Kerr black holes with synchronised scalar hair and boson stars in the Einstein-Friedberg-Lee-Sirlin model