Staticity versus uniqueness in non-rotating black holes with non-minimally self-coupled scalar fields

Non-minimally coupled (\zeta \neq 0) self-interacting scalar field distributions on the domain of outer communications of a non-rotating strictly stationary black hole are studied. It is concluded that for distributions bounded at the horizon the space-time is static, or the field takes the unique values \Phi =\pm 1/\sqrt{\kappa \,\zeta }