  Discrete-Gauss states are a new class of gaussian solutions of the free
Schr\"odinger equation owning discrete rotational symmetry. They are obtained
by acting with a discrete deformation operator onto Laguerre-Gauss modes. We
present a general analytical construction of these states and show the
necessary and sufficient condition for them to host embedded dark beams
structures. We unveil the intimate connection between discrete rotational
symmetry, orbital angular momentum, and the generation of focussing dark beams.
The distinguishing features of focussing dark beams are discussed. The
potential applications of Discrete-Gauss states in advanced optical trapping
and quantum information processing are also briefly discussed.
