Non-integrability Criterium for Normal Variational Equations around an   integrable Subsystem and an example: The Wilbeforce spring-pendulum
  In this paper we analyze the non-integrability of the Wilbeforce pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability. 
Primitivo B. Acosta-Hum\'anez, Martha Alvarez--Ram\'irez, David   Bl\'azquez-Sanz and Joaqu\'in Delgado
math.DS math-ph math.MP