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We address the following question: let F:(R^2,0)->(R^2,0) be an analyticlocal diffeomorphism defined in the neighborhood of the non resonant ellipticfixed point 0 and let \Phi be a formal conjugacy to a normal form N. SupposingF leaves invariant the foliation by circles centered at 0, what is the analyticnature of \Phi and N?

Elliptic fixed points with an invariant foliation: Some facts and more questions