The paper is dealing with interrelations between Cauchy-Riemann structures and Lorentzian geometry. First, the author studies the concept of CR-submanifold [see the reviewer, Geometry of CR-submanifolds (Dordrecht, 1986; Zbl 0605.53001)] in a Lorentz manifold. He proves that a totally umbilical CR-submanifold of a Lorentz manifold is not necessarily totally geodesic provided the holomorphic distribution is light-like. Then he introduces and studies the concept of affine conformal motion and obtains characterizations of Lorentzian manifolds admitting an infinitesimal affine conformal motion.