Characterization of real-analytic infinitesimal CR automorphisms for a class of hypersurfaces in ${\mathbb{C}}^4\mathrm{.}$

In this paper, motivated by the work of Kim and Kolar for the case of pseudoconvex models which are sums of squares of polynomials, we study the Lie algebra of real-analytic infinitesimal $CR$ automorphisms of a model hypersurface $M_0$ given by \begin{equation} M_0= {(z,w) \in \mathbb C^{3} \times \mathbb C \ | \ \Im w= P\bar Q + Q\bar P + R\bar R }, \end{equation} where $P,$ $Q$ and $R$ are homogeneous polynomials. In particular, we classify $M_0$ with respect to the description of its nilpotent rotations when $P,$ $Q$ and $R$ are monomials. We also give an example of a model $M_0$ for which the real dimension of its generalized (exotic) rotations is $3.$