Painlevé analysis and integrability of the trapped ionic system

Integrability in the Painlevé sense of the trapped ionic system in the quadrupole field with superpositions of rotationally symmetric hexapole and octopole fields is studied. Five integrable cases of the system are reported. First Integrals of the planar motion are founded. Confirming three-dimensional integrability of the equations of motion, the third explicit integrals of motion are constructed directly for each case. We carried out a numerical study to observe the regularity and chaotic regions via the Poincaré surface of sections, and corroborate the analytical results.