The authors study the generalized Henon-Heiles system by systematic exploration and analysis of the Birkhoff-Gustavson normal form and the associated formal integral of motion up to and including the 14th order, which is one order higher than published before. At low energies the formal integral of motion is still an excellent approximation to the exact integral. The convergence properties of the formal integral have been analysed in regular and irregular regions. Strictly convergent behaviour is found in some regions of chaotic motion. No obvious example of divergent behaviour is found. Regions of strict convergence correspond either to regular motion, or to weakly unstable chaotic motion with short-time clustering, characterized by a small value of the finite-time analogue of the Lyapunov exponent.