Elliptic Fixed Points with an Invariant Foliation: Some Facts and More Questions

We address the following question: let F:(ℝ^2,0)→(ℝ^2,0) be an analytic local diffeomorphism defined in the neighborhood of the nonresonant elliptic fixed point 0 and let Φ be a formal conjugacy to a normal form N . Supposing F leaves invariant the foliation by circles centered at 0 , what is the analytic nature of Φ and N ?