  Recently, spherical and static flat space solitons (balls) and
self-gravitating, everywhere regular, asymptotically flat solitons (stars) were
constructed in an Einstein-Proca-Higgs model [1], where a complex vector field
gains mass by coupling to a real scalar field with a Higgs-type potential. The
Proca-Higgs model serves as a UV completion of a complex Proca model with
self-interactions. Here, we construct and examine the mathematical and physical
properties of rotating configurations. In particular, rotation allows
horizon-bearing solutions, including stationary clouds surrounding Kerr black
holes and their non-linear continuation into black holes with Proca-Higgs hair.
