Integrability of the Trapped Ionic System

In this paper we will explore the 2D system 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's and Ziglin-Morales-Ruiz-Ramis's classical methods for the proofs. The monodromy group of the normal variational equation (NVE), which is a Heun equation, is studied for commutativity. Some classical and modern results from the theory of transcendental numbers are used for the research.