We show that three types of hexagonal Zn–Mg–rare-earth (RE) ternary compound are formed in the vicinity of Zn6Mg3RE1. An electron diffraction study revealed that these phases are related with a lattice parameters by a ratio 3:5:7 but with approximately the same c parameters. A preliminary structural model for the phase with medium a parameter has been proposed by high-resolution electron microscopy, based on the structures of the other two phases recently determined. It is shown that the two-dimensional hexagonal superstructures can be described by two kinds of Zn–Mg clusters A and B, linked by RE atoms.