Abstract: Real hyperquadrics in complex space can be locally characterized as the Levi non-degenerate hypersurfaces that possess non-linearizable CR automorphisms. The main step in the proof of such a result is to show that any local CR automorphism of a non-quadratic CR manifold is determined by a part of its 1-jet. A similar result is true for so-called hyperbolic CR manifolds of CR dimension two and codimension two (here one has to exclude direct products with a quadratic factor) and for CR manifolds with Levi form in general position in CR dimension n and codimension k with 2 < k < n²-2 (in this case even quadric automorphisms are determined by its 1-jet). Though, it was believed that this is a general phenomenon the problem was open for elliptic CR manifolds. Surprisingly, it turned out that there do exist elliptic CR manifolds with non-linearizable CR automorphisms that are not related to quadrics. In joint work with V. Ezhov the author found a complete description of these manifolds based on a correspondence between so-called torsion-free CR manifolds and second order ODE.