Transcendental holomorphic maps between real algebraic manifolds in a complex space

We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.