  The phase diagram of the spin-1 Ising model in the presence of a biaxial
crystal-field anisotropy is studied within the framework of a variational
approach, based on the Bogolyubov inequality for the free energy. We have
investigated the effects of a transverse crystal-field $D_{y}$ on the phase
diagram in the $T-D_{x}$ plane. Results obtained by using effective-field
theory (EFT) on the honeycomb (% $z=3$), square ($z=4$) and simple cubic
($z=6$) lattices ($z$ is the coordination number) show only continuous phase
transitions, while the variational approach presents first-order and continuous
phase transitions for $D_{y}=0$% . We have also used the EFT for larger values
of $z$ and we observe the presence of tricritical points in the phase diagrams,
for $z\geq 7$, in accordance with the variational approach results.
