  In this work, we look at the symmetry of normal modes in symmetric
structures, particularly structures with cyclic symmetry. We show that normal
modes of symmetric structures have different levels of symmetry, or
symmetricity. One novel theoretical result of this work is that, for a ring
structure with $m$ subunits, the symmetricity of the normal modes falls into
$m$ groups of equal size, with normal modes in each group having the same
symmetricity. The normal modes in each group can be computed separately, using
a much smaller amount of memory and time (up to $m^3$ less), thus making it
applicable to larger complexes. We show that normal modes with perfect symmetry
or anti-symmetry have no degeneracy while the rest of the modes have a
degeneracy of two. We show also how symmetry in normal modes correlates with
symmetry in structure. While a broken symmetry in structure generally leads to
a loss of symmetricity in symmetric normal modes, the symmetricity of some
symmetric normal modes is preserved even when symmetry in structure is broken.
This work suggests a deeper reason for the existence of symmetric complexes:
that they may be formed not only for structural purpose, but likely also for a
dynamical reason, that certain symmetry is needed by a given complex to obtain
certain symmetric motions that are functionally critical.
