  The aim of this paper is to describe the geometry of conformal structures in
Lorentzian signature, which admit a lightlike conformal Killing vector field
whose corresponding adjoint tractor acts as complex structure on the standard
tractor bundle of conformal geometry. Key to the treatment of this problem is
CR-geometry and the Fefferman construction. In fact, we will consider here
partially integrable CR-structures and a slightly generalised Fefferman
construction for these, which we call the $\ell$-Fefferman construction. We
show that a certain class of $\ell$-Fefferman metrics on partially integrable
CR-spaces provides all solutions to our problem concerning the complex
structures.
