Blow-ups and infinitesimal automorphisms of CR-manifolds

For a real-analytic connected CR-hypersurface M of CR-dimension n >= 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra s = s(M): (i) either dim s = n(2) + 4n + 3 and M is spherical everywhere; (ii) or dim s <= n(2) + 2n + 2 + delta(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.