A master equation for Carter-separable stationary axisymmetric spacetimes and compatible sources

We show that the remaining diagonal Einstein equations in the Carter-projective sector of stationary axisymmetric spacetimes are equivalent to a single sourced master equation. The projective structure is taken as the input fixed by the off-diagonal Einstein equations. In the anti-aligned exponential branch, which contains the Kerr–Carter and Plebański–Demiański real section, the remaining diagonal Einstein system reduces to $${\mathcal{L}}_{\mathrm{C}\mathrm{P}}[\mathrm{\Delta },Y]=16\pi \mathrm{\Sigma }(T_{\widehat{0}\widehat{0}}+T_{\widehat{3}\widehat{3}}),$$ where $\mathrm{\Delta }(r)$ and $Y(x)$ are the radial and angular structure functions. The reduction is accompanied by two geometric diagonal identities of the Einstein tensor, which become algebraic compatibility conditions on admissible matter sources. In the homogeneous limit, the vacuum–$\mathrm{\Lambda }$ Kerr–Carter and Plebański–Demiański families are recovered as solutions of the same master operator. We also show the projective covariance of the construction and discuss compatible sources, including the aligned Maxwell field and separable anisotropic examples.