Conformal structures with $G_2$-symmetric twistor distribution

For any 4D split-signature conformal structure, there is an induced twistor distribution on the 5D space of all self-dual totally null 2-planes, which is (2, 3, 5) when the conformal structure is not anti-self-dual. Several examples where the twistor distribution achieves maximal symmetry (the split-real form of the exceptional simple Lie algebra of type $$\textrm{G}_2$$ G 2 ) were previously known, and these include fascinating examples arising from the rolling of surfaces without twisting or slipping. We establish a complete local classification result for achieving maximal symmetry of the twistor distribution, identified among those homogeneous 4D split-conformal structures for which the conformal symmetry algebra induces a multiply-transitive action on the 5D space. Furthermore, we discuss geometric properties of these conformal structures such as their curvature, holonomy, and existence of Einstein representatives.