  The aim of this article is twofold: First we 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. We find a canonical formal normal
form and provide a complete analytic classification (in formal generic cases)
in terms of a collection of functional invariants. Their restriction to an
irreductible component of the zero locus of the first integral reduces to the
Birkhoff--\'Ecalle--Voronin modulus of the 1-dimensional restricted parabolic
germ.
  We then generalize this classification also to germs of anti-holomorphic
diffeomorphisms of $(\mathbb{C}^2,0)$ whose square iterate is of the above
form.
  Related to it, we solve the problem of both formal and analytic
classification of germs of real analytic surfaces in $\mathbb{C}^2$ with
non-degenerate 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 hyperplane of $\mathbb{C}^2$.
