A real analytic diffeomorphism admitting a non resonant elliptic fixed point is always formally conjugated to a formal integrable system, its Birkhoff Normal Form (BNF) but is not in general analytically of even topologically conjugated to an integrable system. I will address in this context the following questions: Is the BNF generally convergent or divergent? What are the dynamical consequences of the convergence of a formal object like the BNF, in particular does it imply integrability? Can one perturb a real analytic symplectomorphism so that it becomes integrable in a neighborhood of the origin?