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Existence of Real Analytic Surfaces with Hyperbolic Complex Tangent that Are Formally but Not Holomorphically Equivalent to Quadrics

We shall prove the existence of a real analytic sur- face in C 2 that is formally but not holomorphically equivalent to a quadratic surface with a hyperbolic complex tangent. The result is obtained through the pair of holomorphic involutions of Moser-Webster for a real analytic surface with a complex tan- gent of non-vanishing Bishop invariant. Our method is also used to show the existence of real analytic reversible maps of the real plane, defined near the origin, that are formally, but not real an- alytically, equivalent to a linear rotation.