Quasi integral of motion for axisymmetric potentials

We present an estimate of the third integral of motion for axisymmetric three-dimensional potentials. This estimate is based on a Staeckel approximation and is explicitly written as a function of the potential. We tested this scheme for the Besancon Galactic model and two other disc-halo models and find that orbits of disc stars have an accurately conserved third quasi integral. The accuracy ranges from of 0.1% to 1% for heights varying from z = 0~kpc to z= 6 kpc and Galactocentric radii R from 5 to 15kpc. We also tested the usefulness of this quasi integral in analytic distribution functions of disc stellar populations: we show that the distribution function remains approximately stationary and that it allows to recover the potential and forces by applying Jeans equations to its moments.