Analysis of rotation-vibration relative equilibria on the example of a tetrahedral four atom molecule K. Efstathiou, D. A. Sadovskií and B. I. Zhilinskií Université du Littoral, UMR 8101 du CNRS, 59140 Dunkerque, France. SIADS (SIAM Journal of Applied Dynamical Systems) ms\#060001 submitted 23 January 2003, accepted 3 October 2003. Supported by the EU project Mechanics and Symmetry in Europe (MASIE), contract HPRN-CT-2000-00113. Abstract. We study relative equilibria (RE) of a nonrigid molecule which vibrates about a well defined equilibrium configuration and rotates as a whole. Our analysis unifies the theory of rotational and vibrational RE. We rely on the detailed study of the symmetry group action on the initial and reduced phase space of our system and consider the consequences of this action for the dynamics of the system. We develop our approach on the concrete example of a four-atomic molecule A4 with tetrahedral equilibrium configuration, a dynamical system with six vibrational degrees of freedom. Further applications and illustrations of our results can be found in [Van Hecke, et al, Eur. Phys. J. D 17, 13 (2001)] Keywords. small vibrations, vibration-rotation of molecules, spherical top, relative equilibria, 1:1:1 resonant oscillator, normalization, reduction, bifurcations, orbit space, finite group action, reversing symmetry, Molien generating function, integrity basis AMS subject classifications. 37J15, 37J35, 37J40, 81V55, 58D19