  In this paper we analyze nonlinear dynamics of the fullerene molecule. We
prove the existence of global branches of periodic solutions emerging from an
icosahedral equilibrium (nonlinear normal modes). We also determine the
symmetric properties of the branches of nonlinear normal modes for maximal
orbit types. We find several solutions which are standing and rotating waves
that propagate along the molecule with icosahedral, tetrahedral, pentagonal and
triangular symmetries. We complement our theoretical results with numerical
computations using Newton's method.
