Abstract
For a real-analytic connected CR-hypersurface M of CR-dimension $$n\geqslant 1$$
n
⩾
1
having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra $$\mathfrak {s}={\mathfrak {s}}(M)$$
s
=
s
(
M
)
: (i) either $$\dim {\mathfrak {s}}=n^2+4n+3$$
dim
s
=
n
2
+
4
n
+
3
and M is spherical everywhere; (ii) or $$\dim {\mathfrak {s}}\leqslant n^2+2n+2+\delta _{2,n}$$
dim
s
⩽
n
2
+
2
n
+
2
+
δ
2
,
n
and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. 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.