  In our recent work [25] we showed that $C^\infty$ CR-diffeomorphisms of
real-analytic Levi-nonflat hypersurfaces in $\mathbb C^{2}$ are not analytic in
general. This result raised again the question on the nature of CR-maps of
real-analytic hypersurfaces.
  In this paper, we give a complete picture of what CR-maps actually are.
First, we discover an analytic continuation phenomenon for CR-diffeomorphisms
which we call the {\em sectorial analyticity property}. It appears to be the
optimal regularity property for CR-diffeomorphisms in general. We emphasize
that such type of extension never appeared previously in the literature.
Second, we introduce the class of {\em Fuchsian type hypersurfaces} and prove
that (infinitesimal generators of) CR-automorphisms of a Fuchsian type
hypersurface are still analytic. In particular, this solves a problem
formulated in [28].
  Finally, we prove a regularity result for {\em formal} CR-automorphisms of
Fuchsian type hypersurfaces.
