Abstract
A new sufficient condition to prove non-integrability of Hamiltonian systems with symmetric variational equations is proposed. Though the present condition is taken as an extension of Ziglin and Yoshida's “non-resonant condition”, the algorithmic bottle-neck proving non-integrability in high dimensional cases can be removed here. As a result, the non-existence of additional integrals of various symmetric Hamiltonian systems with an arbitrary number of degrees of freedom is proven.
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