We analyze a renormalization group transformation $\mathcal R$ for partially analytic
Hamiltonians, with emphasis on what seems to be needed for the construction
of non-integrable fixed points. Under certain assumptions, which are supported by numerical
data in the golden mean case, we prove that such a fixed point has a critical
invariant torus. The proof is constructive and can be used for numerical computations.
We also relate $\mathcal R$ to a renormalization group transformation for commuting maps.