  We provide a geometric framework for the construction of non-vacuum black
holes whose metrics are stationary and axisymmetric. Under suitable assumptions
we show that the Einstein equations reduce to an Einstein-harmonic map type
system and analyze the compatibility of the resulting equations.
  This framework will be fundamental to our forthcoming construction of
metric-stationary axisymmetric bifurcations of Kerr solving the
Einstein--Klein--Gordon system, and as such, we include specializations of all
of our formulas to the case of a time-periodic massive scalar field.
