  A study of the $p d \to p d \eta$ reaction in the energy range where the
recent data from Uppsala are available, is done in the two-step model of $\eta$
production including the final state interaction. The $\eta -d$ final state
interaction is incorporated through the solution of the Lippmann Schwinger
equation using an elastic scattering matrix element, $T_{\eta d \to \eta d}$,
which is required to be half off-shell. It is written in a factorized form,
with an off-shell form factor multiplying an on-shell part given by an
effective range expansion up to the fourth power in momentum. The parameters of
this expansion have been taken from an existing recent relativistic Faddeev
equation solution for the $\eta NN$ system corresponding to different $\eta-N$
scattering amplitudes. Calculations have also been done using few body
equations within a finite rank approximation (FRA) to generate $T_{\eta d \to
\eta d}$. The $p-d$ final state interaction is included in the spirit of the
Watson-Migdal prescription by multiplying the matrix element by the inverse of
the Jost function. The $\eta-d$ interaction is found to be dominant in the
region of small invariant $\eta -d$ mass, $M_{\eta d}$. The $p-d$ interaction
enhances the cross section in the whole region of $M_{\eta d}$, but is larger
for large $M_{\eta d}$. We find nearly isotropic angular distributions of the
proton and the deuteron in the final state. All the above observations are in
agreement with data. The production mechanism for the entire range of the
existing data on the $p d \to p d \eta$ reaction seems to be dominated by the
two-step model of $\eta$ production.
