For the Integrability of the 2D Trapped Ionic System

In this paper we will explore the 2D system with Hamiltonian H=12(pr2+pz2)+Ar2+Bz2+Cz3+Dr2z+Ez4+Fr2z2+Gr4, describing trapped ionic system in the quadrapole field with a superposition of rationally symmetric hexapole and octopole fields for meromorphic integrability. We use the Lyapunov and Ziglin-Morales-Ruiz-Ramis’s classical methods for the proofs.