In this joint work with Martin Klimes, we solve the problem of both formal and analytic classification of germs of real analytic surfaces in $(\mathbb{C}^2$ with CR singularities of exceptional hyperbolic type, under the assumption that the surface is holomorphically flat, i.e. that it can be locally holomorphically embedded in a real hypersurface of $(\mathbb{C}^2$. It happens that this relies on the study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin.
