Painlevé analysis and integrability of three-dimensional Armbruster Guckenheimer Kim galactic potential

The three-dimensional Armbruster Guckenheimer Kim galactic potential in the general form is considered. The integrability of this problem is performed by using the Painlevé analysis. We report three cases of integrability, exact integrals of motion are obtained explicitly for each case. A numerical experiment is provided to investigate the classical phase space and explore chaos-order-chaos phenomenon.