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

We address the following question: let $F:(\mathbb {R}^2,0)\to(\mathbb {R}^2,0)$ be an analytic local diffeomorphism defined in the neighborhood of the nonresonant elliptic fixed point 0 and let $\Phi$ 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 $\Phi$ and $N$?