A CR structure on a real three manifold , with local coordinates , is given intrinsically by equivalence classes of one-forms, one real, one complex and its complex conjugate . If we denote the real one-form by and the complex one-form by , then these are defined up to the transformations::
Any three-manifold with a CR structure is referred to as a three-dimensional CR manifold. There are special classes (referred to as embeddable) of three-dimensional CR manifolds that can be directly embedded into .
We show how the choice of any specific asymptotically shear-free NGC induces a CR structure on . Though there are several ways of arriving at this CR structure, the simplest way is to look at the asymptotic null tetrad system associated with the asymptotically shear-free NGC, i.e., look at the (, , , ) of Equation (274). The associated dual one-forms, restricted to (after a conformal rescaling of ), become (with a slight notational dishonesty),
The dual vectors – also describing the CR structure – are
Therefore, for the situation discussed here, where we have singled out a unique asymptotically shear-free NGC and associated complex world line, we have a uniquely chosen CR structure induced on .
To see how our three manifold, , can be imbedded into we introduce the CR equation 
and seek two independent (complex) solutions, that define the embedding of into with coordinates .
We have immediately that is a solution. The second solution is also easily found; we see directly from Equation (175) ,
the inverse to , is a CR function and that we can consider to be embedded in the of .
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