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In this paper we analyze the non-integrability of the Wilbeforce pendulum bymeans of Morales-Ramis theory in where is enough to prove that the Galois groupof the variational equation is not virtually abelian. We obtain thesenon-integrability results due to the algebrization of the variational equationfalls into a Heun differential equation with four singularities and then weapply Kovacic's algorithm to determine its non-integrability.

Non-integrability Criterium for Normal Variational Equations around an integrable Subsystem and an example: The Wilbeforce spring-pendulum