Intra- and interpolyad structures are investigated in the vibrational energy pattern of acetylene, using the spectroscopic Hamiltonian presented in previous papers in this series [see El Idrissi , J. Chem. Phys. 110, 2074 (1999)]. The existence of two constants of the motion is shown to generate very regular patterns in the manifold of vibrational energy levels. Distinct regular and oscillatory contributions are evidenced in the number of vibrational levels in the main polyads, which are fully reproduced using the generating function presented in Sadovskii and Zhilinskii [J. Chem. Phys. 103, 10520 (1995)]. Further developments of this approach are outlined.