Bushes of normal modes represent the exact excitations in the nonlinear physical systems with discrete symmetries [Physica D 117 (1998) 43]. The present paper is the continuation of our previous paper [Physica D 166 (2002) 208], where these dynamical objects of new type were discussed for the monoatomic nonlinear chains. Here, we develop a simple crystallographic method for finding bushes in nonlinear chains and investigate stability of one-dimensional and two-dimensional vibrational bushes for both FPU-α and FPU-β models, in particular, of those revealed recently in [Physica D 175 (2003) 31].

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All possible symmetry-determined nonlinear normal modes (also called simple periodic orbits, one-mode solutions, etc.) in both hard and soft Fermi-Pasta-Ulam β chains are discussed. A general method for studying their stability in the thermodynamic limit as well as its application for each of the above nonlinear normal modes are presented.

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In-plane nonlinear delocalized vibrations in uniformly stretched single-layer graphene (space group P6mm) are considered with the aid of the group-theoretical methods. These methods were developed by authors earlier in the framework of the theory of the bushes of nonlinear normal modes (NNMs). We have found that only 4 symmetry-determined NNMs (one-dimensional bushes), as well as 14 twodimensional, 1 three-dimensional and 6 four-dimensional vibrational bushes are possible in graphene. They are exact solutions to the dynamical equations of this two-dimensional crystal. Prospects of further research are discussed.