Degenerate hypersurfaces with a two-parametric family of automorphisms

We give a complete classification of Levi-degenerate hypersurfaces of finite type in ℂ² with two-dimensional symmetry groups. Our analysis is based on the classification of two-dimensional Lie algebras and an explicit description of isotropy groups for such hypersurfaces, which follows from the construction of Chern–Moser type normal forms at points of finite type, developed in [M. Kolář, 'Normal forms for hypersurfaces of finite type in ℂ²', Math. Res. Lett. 12 (2005), pp. 897–910].