The conformal-to-Carter metric is the unique ansatz for the stated geometric conditions, and in Einstein-Maxwell theory the only non-trivial solutions are the Ovcharenko-Podolsky class for fully non-aligned non-null EM fields and the Plebański-Demiański class for double-aligned non-null EM fields.
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Uniqueness of stationary axisymmetric type D black holes with non-aligned electromagnetic field
The conformal-to-Carter metric is the unique ansatz for the stated geometric conditions, and in Einstein-Maxwell theory the only non-trivial solutions are the Ovcharenko-Podolsky class for fully non-aligned non-null EM fields and the Plebański-Demiański class for double-aligned non-null EM fields.