  Coherent photoproduction of the neutral pion and eta meson on the deuteron,
$\gamma{d}${$\to$}$\pi^0\eta{d}$, has been experimentally studied at incident
photon energies ranging from the reaction threshold to 1.15 GeV. The total
cross section demonstrates a rapid rise below 1 GeV. The data are
underestimated by the existing theoretical calculations based on quasi-free
$\pi^0 \eta$ photoproduction on the nucleon followed by deuteron coalescence.
At the same time, the data are rather well reproduced by the calculations
taking into account the final-state interaction. We have also measured for the
first time the differential cross sections: the $\pi^0 \eta$ invariant-mass
distribution $d\sigma/dM_{\pi \eta}$, the $\pi^0 d$ invariant-mass distribution
$d\sigma/dM_{\pi d}$, the $\eta d$ invariant-mass distribution
$d\sigma/dM_{\eta d}$, and the distribution over the deuteron emission angle
$d\sigma/d\Omega_d$ in the overall center-of-mass frame. The measured cross
section $d\sigma/d\Omega_d$ does not exhibit strongly backward-peaking behavior
predicted by the calculations. At all incident energies, an increase in
$d\sigma/dM_{\eta d}$ near the $\eta d$ threshold is observed, which indicates
a bound or virtual $\eta d$ state resulting from a strong attraction between
$\eta$ and a deuteron. The possibilities of using coherent $\pi^0\eta$
photoproduction on a nucleus to study the $\eta$-nuclear interaction are also
discussed.
