Algebraic non-integrability of the Cohen map

The map φ(x, y) = (√1 + x² - y, x) of the plane is area preserving and has the remarkable property that in numerical studies it shows exact integrability: The plane is a union of smooth, disjoint, invariant curves of the map φ. However, the integral has not explicitly been known. In the current paper we will show that the map φ does not have an algebraic integral, i.e., there is no non-constant function F(x, y) such that 1. F ○ φ = F; 2. There exists a polynomial G(x, y, z) of three variables with G(x, y, F(x, y)) = 0. Thus, the integral of φ, if it does exist, will have complicated singularities. We also argue that if there is an analytic integral F, then there would be a dense set of its level curves which are algebraic, and an uncountable and dense set of its level curves which are not algebraic.