Non-integrability of a Hamiltonian system and Legendre functions

Abstract: We investigate the solvability of the Galois group of the associated Legendre equation and we apply it it for study integrability to a Hamiltonian system with a homogeneous potential of degree 6. In this paper, we study the Hamiltonian system with Hamiltonian \ $H=\frac{1}{2}(p_r^2+p_z^2)+r^6+Ar^2z^4+Dr^3z^3+Br^4z^2+Cz^6$, ($A,\, B,\, C,\, D \in \mathbb{R}$) for meromorphic integrability. The technique is an application of the Ziglin-Moralez-Ruiz-Ramis-Simo Theory.