Integrals of motion for non-axisymmetric potentials

Context. The modelling of stationary galactic stellar populations can be performed using distribution functions. Aims. This paper aims to write explicit integrals of motion and distribution functions. Methods. We propose an analytic formulation of the integrals of motion with an explicit dependence on potential. This formulation applies to potentials with rotational symmetry or triaxial symmetry. It is exact for Stackel potentials and approximate for other potentials. Results. Modelling a stationary stellar population using these integrals of motion allows the force field to be found with satisfactory accuracy. On the other hand, the mass density distribution that generates the force field and the gravitational potential is recovered with less accuracy due to lower precision in modelling box-type orbits.