Stationary axisymmetric black holes with matter

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 Einstein equations reduce to an Einstein-harmonic map type system analyze compatibility resulting equations. This will be fundamental our forthcoming metric-stationary axisymmetric bifurcations Kerr solving Einstein--Klein--Gordon system, as such, include specializations all formulas case time-periodic massive scalar field.