Finding conical intersections between potential energy surfaces is key to understanding photochemistry, particular for ultrafast non-adiabatic reactions. An earlier post pointed out how often these conical intersections occur at a molecular geometry where there is a local triangular symmetry. This leads naturally to an effective Hamiltonian which has a C_3 (or higher) symmetry and the degenerate eigenstates are in the two-dimensional E representation.
However, there is more to the story...
There is a nice recent review Electronically excited states and photodynamics: a continuing challenge by Plasser, Barabatti, Aquino, and Lischka.
They present a Table of common motifs for "primitive conical intersections".
The authors also suggest there are three distinct excited state pathways, summarised schematically in the diagram below. They respectively occur in the molecules shown: