# Integrals of motion for non-axisymmetric potentials

## TL;DR

This paper develops explicit integrals of motion for non-axisymmetric galactic potentials, enabling better modeling of stellar populations and gravitational fields with a focus on triaxial and Stäckel potentials.

## Contribution

It introduces an analytic formulation of integrals of motion with explicit potential dependence applicable to non-axisymmetric potentials, including approximate solutions for general cases.

## Key findings

- Accurately models stationary stellar populations using derived integrals.
- Recovers force fields with satisfactory accuracy.
- Less precise in reconstructing mass density distributions.

## Abstract

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 St\"ackel 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.