  Let $M$ be a real-analytic connected CR-hypersurface of CR-dimension $n>0$
having a point of Levi-nondegeneracy. The following alternative is demonstrated
for both the symmetry algebra $s$ and the automorphism group $G$ of $M$. Denote
by $d$ the dimension of $s$ or $G$. Then (i) either $d=n^2+4n+3$ and $M$ is
spherical everywhere; (ii) or $d\le n^2+2n+2+\delta_{2,n}$ and in the case of
equality $M$ is spherical of fixed Levi signature in the open dense subset of
Levi-nondegenerate points.
  Explicit examples of CR-hypersurfaces and their infinitesimal and global
automorphisms realizing the bound in (ii) are constructed. We provide many
other models with large symmetry using the technique of blow-up, in particular
we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as
a symmetry.
