**Abstract**

Current Milky Way dynamical models based on actions and angles variables still often rely on the assumptions of a smooth time-independent and axisymmetric gravitational potential. In this talk, I will display analytical forms of distribution functions in angle-action coordinates, representing the thin disc and taking into account the response to a spiral potential through the linearized Boltzmann equation. Various effects such as non-zero mean vertical and radial stellar motions are then predicted. Such a linearized treatment however leads to singularities which prevents from making a quantitative comparison with data at resonances. To capture the behaviour of the distribution function at a resonance, one can average the Hamiltonian over fast variables and re-express the distribution function in terms of a new set of canonical action and angle variables valid in the resonant region. Following the prescription of Binney (2016), one can then assign to the resonant DF the time-average along the orbits of the axisymmetric DF expressed in the new set of actions and angles. This opens the way to quantitatively fitting the detailed effects of the bar and spirals to Gaia data in terms of action-based distribution functions.