A theoretical study of the lowest singlet and triplet potential surfaces (PES) of the BeC2H2 and MgC2H2 systems has been carried out. In both cases a linear singlet state HXCCH (1-SIGMA-+) (X being Be or Mg) is found to be the ground state. We have computed harmonic vibrational frequencies for the different species, which may help to identify some stable adducts. Special attention is paid to the interaction of Be and Mg in their lowest lying 1S, 3P, and 1P states with the pi bond of acetylene. In particular, we have found that the 1A1 PES exhibits a double-well structure for both systems. Crossings between the 1A1 and 1B2 states, which cause conical intersections, have been found, and as a consequence the interaction of X (1P) with acetylene may result in X (1S) + HCCH (1-SIGMA-g+). This is a possible mechanism for the gas-phase quenching of Mg (1P) atoms with acetylene. There are also crossing points between the 3B2 and 1A1 surfaces. Therefore, if conveniently stabilized, such as in matrix isolation conditions, the XC2H2 (3B2) adduct could undergo intersystem crossing to the 1A1 state. Interactions of Be and Mg with the C-H bond, as well as isomerization of cyclic XC2H2 states into X-vinylidene species, have also been studied. It is found that the vinylidene adducts may undergo fragmentation processes quite easily when produced from rearrangement of the cyclic state. This fact, together with the exit channel giving Be (1S) or Mg (1S) and acetylene for the interaction of Be and Mg atoms with acetylene, severely limits the ability of these metals to catalyze acetylene to vinylidene conversion.