  In a case study for integrable PDEs, we construct real analytic, canonical
coordinates for the defocusing NLS equation on the circle, specifically
taylored towards the needs in perturbation theory. They are defined in
neighbourhoods of families of finite dimensional invariant tori and are shown
to satisfy together with their derivatives tame estimates. When expressed in
these coordinates, the dNLS Hamiltonian is in normal form up to order three.
