We study a class of three degree of freedom (3-DOF) Hamiltonian systems that share certain characteristics with the 2-DOF Hénon-Heiles Hamiltonian. 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 (1960 J. Mol. Spectrosc. 5 355) and later by Patterson (1985 J. Chem. Phys. 83 4618) to model triply degenerate vibrations of tetrahedral molecules. We describe relative equilibria (RE) of these systems, classify all qualitatively different family members, and discuss bifurcations of RE involved in the transitions from one region of regular parameter values to the other.