In our paper A Reduction Method for
Higher Order Variational Equations of Hamiltonian Systems, we
illustrate our reduction algorithm by presenting a full reduction for
the classical hard case of the Henon-Heiles system (where a third
variational equation is needed in order to prove non-integrability).
The Maple calculations may be reproduced (and checked) using this file (click here).
The file contains : Maple code used in the calculations, then the
necessary matrices (the variational equations), and then the reduction
process in detail.