A general method for finding all electronic degeneracies lying on the ground-state potential surface of a molecular system is proposed. The method is based on the idea that the spin pairing of the valence electrons is the major factor determining the topology of the potential surface. The number of different spin-pairing arrangements (anchors) that can be constructed from the constituent atoms determines the number of critical points (minima, transition states) on the ground-state surface. It is shown that whereas the interaction between two states leads in general to an avoided crossing of potential surfaces the interactions in a three-state system (consisting of three anchors) lead in general to a 2-fold degeneracy (conical intersection) and in a four-state system to a 3-fold degeneracy. It is further shown that in a 3D world the highest degree of nonaccidental electronic degeneracy is 3. Since the number of anchors in a polyatomic system can be large, in general numerous 3-fold degeneracies exist in the system, independent of nuclear symmetry. The whole topology of the potential surface can be constructed around the degeneracies since minima and transition states are directly accessible from them via a monotonic declining route. A practical procedure for establishing the approximate structures of the 3-fold degenerate "points" and also those of the more familiar 2-fold degeneracies (conical intersections) is proposed.