We demonstrate the uniqueness of the spacetimes recently found by us [Phys. Rev. D 112, 064076 (2025)]. First, we prove that the conformal-to-Carter metric ansatz we used therein is the only one possible for stationary axisymmetric geometries that are of Weyl type D, with geodesic and shear-free principal null directions which are orthogonal to polar directions, and whose specific 1-form $\theta$ is closed. Because this result is general, without employing any field equations, such conformal-to-Carter metric may find interesting applications also in various alternative theories of gravity. Then, we show that in the Einstein-Maxwell theory the only nontrivial electrovacuum solution for the conformal-to-Carter metric with the fully nonaligned and non-null electromagnetic field is the Ovcharenko-Podolský class found in 2025. Complementarily, the only solution with the double-aligned and non-null electromagnetic field is the Plebański-Demiański class found in 1976.

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• Classical black holes
• General relativity
• General relativity equations & solutions