About complex structures in conformal tractor calculus

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.