In this Letter are presented and analyzed conical intersections which appear on the two symmetric sides of the C-2 upsilon line of the C2H molecule. Two conical intersections (CI) of this kind, between the 3 (2)A' and 4 (2)A' electronic states, were found to be only a short distance apart, e.g., similar to0.3 Angstrom for the CC distance of 1.25 Angstrom. It is shown that these two CIs-to be termed CI twins-have opposite "charges" thus forming altogether a weak interaction. By increasing the CC distance, to 1.35 Angstrom, the two twins coalesce to form a single CI. The interaction of this merged pair varies with the distance as q(-1) (as is the case for conical intersections) but, in contrast to ordinary CIs, does not exhibit any topological effects and its intensity is shown to be zero. These features led us to term it as a degenerate CI or concisely DCI. (C) 2001 American Institute of Physics.