The properties of the five-dimensional branching space of conical intersections of three states of the same symmetry (denoted i,j,k) are considered. The results of a perturbative model are compared with multireference configuration interaction calculations for three spectroscopically observed states of the allyl radical. Of particular interest is the three-dimensional subspace of the branching space where two states remain degenerate. The energies, derivative couplings and geometric phase effect are studied in the neighborhood of this degeneracy subspace. The degeneracy subspace includes two kinds of conical intersections, i,j and j,k. The existence of a three-state intersection impacts the phase of the wave functions (and the derivative coupling) traversing a closed loop. For example, in the branching space, the number and kind of conical intersections in a surface bounding the closed loop is constrained if the closed loop contains the three-state intersection.

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