  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.
