Researchers have shown that Einstein's equations for stationary and axisymmetric spacetimes, separable in Carter coordinates, can be reduced to a single master equation. This simplification specifically applies to the diagonal sector of Einstein's equations, assuming that the projective structure is already defined by the non-diagonal equations. This advance is significant because it allows for a unified approach to describing rotating black holes and other complex solutions of general relativity.

The master equation takes the form $\mathcal L_{\rm CP}[\Delta,Y] =16\pi\Sigma\left( T_{\hat0\hat0}+T_{\hat3\hat3} \right)$, where $\Delta(r)$ and $Y(x)$ are functions describing the radial and angular structure of spacetime, respectively. This formulation is particularly relevant in the antialigned exponential branch, which includes the real sections of the Kerr-Carter and Plebański-Demiański metrics. The reduction to this single equation is accompanied by two diagonal geometric identities of the Einstein tensor, which transform into algebraic compatibility conditions for admissible matter sources.

In the homogeneous limit, the well-known families of vacuum solutions with a cosmological constant ($\Lambda$) from Kerr-Carter and Plebański-Demiański are recovered as solutions of this same master operator. This highlights the unifying capability of the new equation. The work also explores the projective covariance of the construction and discusses compatible sources, including aligned Maxwell fields and examples of separable anisotropic sources, which opens new avenues for modeling complex astrophysical systems with greater theoretical coherence.