Normalization of rationally integrable systems

It is well known that any analytic vector field near a singular point admits a normalization à la Poincare-Birkhoff, but this normalization is only formal in general, and the problem of analytic (convergent) normalization is a difficult one. In [26] and [27] we proposed a new approach to the normalization of vector fields, via their intrinsic associated torus actions: an analytic vector field is analytically normalizable near a singular point if and only if its associated torus action is analytic (and not just formal). We then showed that if a vector field is analytically integrable, then its associated torus action is analytic, and therefore the vector field is analytically normalizable [26], [27]. In this paper we extend this analytic normalization result to the case of rationally integrable systems, where the first integrals and commuting vector fields are not required to be analytic, but just rational (that is, quotients of analytic functions or vector fields by analytic functions). For example, any vector field of the type $X = f Y$, where Y is an analytically diagonalizable vector field and f is an analytic function such that $Y (f ) = 0$, is rationally integrable but not necessarily analytically integrable.