  We construct Z_3 vortex solutions in a model in which SU(3) is spontaneously
broken to Z_3. The model is truncated to one in which there are only two
dimensionless free parameters and the interaction of vortices within this
restricted set of models is studied numerically. We find that there is a curve
in the two dimensional space of parameters for which the energy of two
asymptotically separated vortices equals the energy of the vortices at
vanishing separation. This suggests that the inter-vortex potential for Z_3
strings might be flat for these couplings, much like the case of U(1) strings
in the Bogomolnyi limit. However, we argue that the intervortex potential is
attractive at short distances and repulsive at large separations leading to the
possibility of unstable bound states of Z_3 vortices.
