Robinson-Sparling construction of CR structures associated to shearfree null geodesic congruences

We review the construction of Lorentzian metrics, such as Fefferman type metrics, associated to a given 3-dimensional nondegenerate CR manifold $M$, and admitting shearfree null geodesic congruences $N$. This class of metrics is obtained by a lifting procedure from $M$ to $M times mathbb{R}$ devised by I. Robinson and A. Trautman (cf. [71]-[72]) and notably radiative gravitational fields are searched for (cf. e.g. R.K. Sachs, [74]) within the class. Conversely, nondegenerate CR structures arise (by the Robinson-Trautmann construction, [71]) on leaf spaces $mathfrak{M}/N$ associated to space-times $mathfrak{M}$ adapted to given optical structures $((K,L), J)$. The Graham-Sparling construction (cf. [40], [77]) is shown to be a particular case of Robinson-Trautman construction where the complex structure on the complex line bundle $mathrm{Ker}(L)/K to mathfrak{M}$ is induced by an $f$-structure with two complemented frames obtained as a covariant derivative of the given null Killing vector field $N$.