Researchers have developed an exact solution generator for Einstein gravity coupled to nonlinear sigma models in D>3 dimensions. This method is based on Buchdahl transformations, which allow the derivation of scalar-field solutions from vacuum spacetimes. The technique reformulates this mechanism to produce an exact Ricci source, $R_{AB}=C_D(\beta)\partial_A\Sigma\partial_B\Sigma$, where $\Sigma$ is a Buchdahl potential that remains harmonic in the deformed metric.

The process involves identifying, in a connected region of a Ricci-flat seed spacetime, a non-null hypersurface-orthogonal cyclic coordinate with a fixed-sign norm. The logarithm of this norm defines the potential $\Sigma$. A one-parameter redistribution of the cyclic norm then generates the Ricci source. By composing $\Sigma$ with an affinely parametrized geodesic of a non-degenerate target metric, whose squared speed is $C_D(\beta)$, an exact Einstein-sigma model solution is obtained. For positive-definite targets, this implies that the non-trivial scalar map has a one-dimensional image.

This unified generator has been illustrated with various targets (flat, spherical, hyperbolic, and axion-dilaton) and a wide range of seed spacetimes, including Tangherlini, Weyl, Kasner, Rosen-wave, Gowdy, Kaluza-Klein bubble, and C-metric solutions. The construction clearly separates the geometric Buchdahl source from its target-space realization, extending this rank-one cyclic sector across static, cosmological, wave, and accelerating geometries. Furthermore, through the Einstein/Jordan-frame correspondence, the method also provides vacuum tensor-multiscalar solutions.