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We explicitly describe germs of strongly pseudoconvex non-sphericalreal-analytic hypersurfaces $M$ at the origin in $\CC^{n+1}$ for which thegroup of local CR-automorphisms preserving the origin has dimension $d_0(M)$equal to either $n^2-2n+1$ with $n\ge 2$, or $n^2-2n$ with $n\ge 3$. Thedescription is given in terms of equations defining hypersurfaces near theorigin, written in the Chern-Moser normal form. These results are motivated bythe classification of locally homogeneous Levi non-degenerate hypersurfaces in$\CC^3$ with $d_0(M)=1,2$ due to A. Loboda, and complement earlier joint workby V. Ezhov and the author for the case $d_0(M)\ge n^2-2n+2$.

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