In this paper we study infinitesimal CR automorphisms of Levi degenerate hypersurfaces. We illustrate the recent general results of [18], [17], [15], on a class of concrete examples, polynomial models in ${ℂ}^3$ of the form $\Im w=\Re \left(P\left(z\right)\overline{Q\left(z\right)}\right)$, where $P$ and $Q$ are weighted homogeneous holomorphic polynomials in $z=(z_1,z_2)$. We classify such models according to their Lie algebra of infinitesimal CR automorphisms. We also give the first example of a non monomial model which admits a nonlinear rigid automorphism.

We study the Chern-Moser operator for hypersurfaces of finite type in ${ℂ}^2$. Analysing its kernel, we derive explicit results on jet determination for the stability group, and give a description of infinitesimal CR automorphisms of such manifolds.