In this article we report findings regarding various conical intersections between consecutive pairs of the five lowest (2)A(') states of the C2H molecule. We found that conical intersections exist between each two consecutive (2)A(') states. We showed that except for small (high-energy) regions in configuration space, the two lowest adiabatic states (i.e., the 1 (2)A' and the 2 (2)A') form a quasi-isolated system with respect to the higher states. We also revealed the existence of degenerate parabolical intersections, those with a topological (Berry) phase zero, formed by merging two conical intersections belonging to the 3 (2)A' and the 4 (2)A' states, and suggested a Jahn-Teller-type model to analyze them. Finally, we examined the possibility that the "frozen" locations of the carbons can be considered as points of conical intersection. We found that the relevant two-state topological phase is not zero nor a multiple of pi, but that surrounding both carbons yields a zero topological phase.