ON A THREE DEGREE OF FREEDOM GENERALIZATION OF THE TWO DEGREE OF FREEDOM HÉNON-HEILES HAMILTONIAN K. Efstathiou and D. A. Sadovskií Université du Littoral, UMR 8101 du CNRS, 59\,140 Dunkerque, France We study a class of three degrees of freedom (3-DOF) Hamiltonian systems that share certain characteristics with the 2-DOF Hénon-Heiles system. Our systems represent a 1:1:1 resonant three-oscillator whose principal nonlinear perturbation is the cubic potential term $xyz$ with tetrahedral symmetry. After normalizing and reducing the 1:1:1 oscillator symmetry, we show that near the limit of linearization all our systems can be described as a one-parametric family. Such reduced systems have been suggested earlier by Hecht in 1960 and later by Patterson in 1985 to model triply degenerate vibrations of tetrahedral molecules. We describe relative equilibria of these systems, classify all qualitatively different family members, and discuss bifurcations of relative equilibria involved in the transitions from one region of regular parameter values to the other. received 21 May 2003, in final form 26 Sept 2003, published 21 Nov 2003 Math Subject Classification 37J15 Nonlinearity 17, 415-446 (2004) doi:10.1088/0951-7715/17/2/003