A theoretical treatment of the molecular vibration-rotation problem of homonuclear nonlinear triatomic molecules with special attention to O 3 16, is developed. If the molecule does not possess an equilateral equilibrium configuration, there are then three equilibrium configurations corresponding to cyclic permutations of the particles, and during the motion of the molecule there is some probability it will penetrate the barrier separating these configurations. It is necessary to introduce a system of internal coordinates capable of representing all configurations of the molecule, and this has been done in such a way that one coordinate is an angle and cyclic permutation corresponds to the substitution χ → χ + 2π 3 . The Hamiltonian operator in these coordinates and in terms of the irreducible matrix representations of the rotation group is derived on the basis of choosing the standard configuration with the principal axes coincident with the coordinate axes. Accordingly the Hamiltonian operator is invariant under the complete permutation group of the particles. The operators effecting these substitutions are constants of the motion which serve to classify the symmetry properties of the states. The symmetry properties of the states are worked out for O 3 16, one class consisting of even or odd periodic functions of χ of period 2π 3 and connecting only even-even components of the angular momentum, the other class consisting of even or odd antiperiodic functions of χ, i.e., changing sign on χ → χ + 2π 3 and connecting only the odd-odd components of angular momentum. The selection rules are obtained, noteworthy among which is the rigourous rule that transitions can occur only from one class to the other. Due to penetration of the barrier these two classes of states are separated in energy, and for sufficiently high barriers it is shown that the even periodic will lie slightly lower in energy than its corresponding even antiperiodic solution, whereas the reverse is the case for the odd solutions. Although the potential energy function for the normal ozone molecule is not known the paper concludes with some considerations suggesting that the barrier is lower than the dissociation energy (1.04 ev) into the normal oxygen atom and molecule, and lies at χ = π 3 with the other internal coordinates in the neighborhood of their equilibrium values. At this configuration the molecule has the acute isosceles triangular form, the two closest oxygen atoms having a separation close to that of the oxygen molecule.