Do average Hamiltonians exist?

The word "average" and its variations became popular in the sixties and imp licitly carried the idea that "averaging" methods lead to "average" Hamilto nians. However, given the Hamiltonian H = H-0(J) + epsilon R(theta, J), (ep silon < < 1), the problem of transforming it into a new Hamiltonian H* (J*) (dependent only on the new actions J*), through a canonical transformation given by zero-average trigonometrical series has no general solution at or ders higher than the first.