  Generic Hamiltonian systems have a mixed phase space, where classically
disjoint regions of regular and chaotic motion coexist. We present an iterative
method to construct an integrable approximation, which resembles the regular
dynamics of a given mixed system and extends it into the chaotic region. The
method is based on the construction of an integrable approximation in action
representation which is then improved in phase space by iterative applications
of canonical transformations. This method works for strongly perturbed systems
and arbitrary degrees of freedom. We apply it to the standard map and the
cosine billiard.
