  The stability and electronic structure of competing silicene phases under
in-plane compressive stress, either free-standing or on the ZrB$_2$(0001)
surface, has been studied by first-principles calculations. A particular
($\sqrt{3}\times\sqrt{3}$)-reconstructed structural modification was found to
be stable on the ZrB$_2$(0001) surface under epitaxial conditions. In contrast
to the planar and buckled forms of free-standing silicene, in this
"planar-like" phase, all but one of the Si atoms per hexagon reside in a single
plane. While without substrate, for a wide range of strain, this phase is
energetically less favorable than the buckled one, it is calculated to
represent the ground state on the ZrB$_2$(0001) surface. The atomic positions
are found to be determined by the interactions with the nearest neighbor Zr
atoms competing with Si-Si bonding interactions provided by the constraint of
the honeycomb lattice.
