  We obtain spinning boson star solutions and hairy black holes with
synchronised hair in the Einstein-Klein-Gordon model, wherein the scalar field
is massive, complex and with a non-minimal coupling to the Ricci scalar. The
existence of these hairy black holes in this model provides yet another
manifestation of the universality of the synchronisation mechanism to endow
spinning black holes with hair. We study the variation of the physical
properties of the boson stars and hairy black holes with the coupling parameter
between the scalar field and the curvature, showing that they are,
qualitatively, identical to those in the minimally coupled case. By discussing
the conformal transformation to the Einstein frame, we argue that the solutions
herein provide new rotating boson star and hairy black hole solutions in the
minimally coupled theory, with a particular potential, and that no spherically
symmetric hairy black hole solutions exist in the non-minimally coupled theory,
under a condition of conformal regularity.
