  The most general form for symmetric modes of nonlinear discrete-symmetry
systems with nonlinearity depending on the modulus of the field is presented.
Vortex solutions are demonstrated to behave as Bloch modes characterized by an
angular Bloch momentum associated to a periodic variable, periodicity being
fixed by the order of discrete point-symmetry of the system. The concept of
angular Bloch momentum is thus introduced to generalize the usual definition of
angular momentum to cases where O(2) -symmetry no longer holds. The
conservation of angular Bloch momentum during propagation is demonstrated.
