Canonical structure of renormalization group equations and separability   of Hamiltonian systems
  We investigate perturbed Hamiltonian systems with two degrees of freedom by renormalization group method, which derives a reduced equation called renormalization group equation (RGE) by handling secular terms. We found that RGE is not always a Hamiltonian system. The necessary and sufficient condition that RGE becomes a Hamiltonian system up to the second leading order of a small parameter is that the original system is separable by Cartesian coordinates.   Moreover, RGE is integrable when it is a Hamiltonian system. These statements are partial generalizations of our previous paper. 
Yoshiyuki Y. Yamaguchi and Yasusada Nambu