On Chern-Moser normal forms of strongly pseudoconvex hypersurfaces with high-dimensional stability group

We explicitly describe germs of strongly pseudoconvex nonspherical realanalytic hypersurfaces M at the origin in Cⁿ⁺¹ for which the group of local CR-automorphisms preserving the origin has dimension d₀(M) equal to either n² - 2n + 1 with n ≥ 2 or n² - 2n with n ≥ 3. The description is given in terms of equations defining hypersurfaces near the origin, which are written in the Chern-Moser normal form. These results are motivated by the classification of locally homogeneous Levi nondegenerate hypersurfaces in C³ with d₀(M) = 1, 2 due to A. Loboda, and they complement earlier joint work by V. Ezhov and the author for the case d₀(M) ≥ n² - 2n + 2.