  The zero-temperature ground-state (GS) properties and phase diagram of a
frustrated spin-1 $J_{1}$--$J_{2}$ Heisenberg model on the checkerboard square
lattice are studied, using the coupled cluster method. We consider the case
where the nearest-neighbour exchange bonds have strength $J_{1}>0$ and the
next-nearest-neighbour exchange bonds present (viz., in the checkerboard
pattern of the planar pyrochlore) have strength $J_{2} \equiv \kappa J_{1}>0$.
We find significant differences from both the spin-1/2 and classical versions
of the model. We find that the spin-1 model has a first phase transition at
$\kappa_{c_{1}} \approx 1.00 \pm 0.01$ (as does the classical model at
$\kappa_{{\rm cl}}=1$) between two antiferromagnetic phases, viz., a
quasiclassical N\'{e}el phase (for $\kappa < \kappa_{c_{1}}$) and one of the
infinitely degenerate family of quasiclassical phases (for $\kappa >
\kappa_{c_{1}}$) that exists in the classical model for $\kappa > \kappa_{{\rm
cl}}$, which is now chosen by the {\it order by disorder} mechanism as
(probably) the "doubled N\'{e}el" (or N\'{e}el$^{\ast}$) state. By contrast,
none of this family survives quantum fluctuations to form a stable GS phase in
the spin-1/2 case. We also find evidence for a second quantum critical point at
$\kappa_{c_{2}} \approx 2.0 \pm 0.5$ in the spin-1 model, such that for $\kappa
> \kappa_{c_{2}}$ the quasiclassical (N\'{e}el$^{\ast}$) ordering melts and a
nonclassical phase appears, which, on the basis of preliminary evidence,
appears unlikely to have crossed-dimer valence-bond crystalline (CDVBC)
ordering, as in the spin-1/2 case. Unlike in the spin-1/2 case, where the
N\'{e}el and CDVBC phases are separated by a phase with plaquette valence-bond
crystalline (PVBC) ordering, we find very preliminary evidence for such a PVBC
state in the spin-1 model for all $\kappa > \kappa_{c_{2}}$.
