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Coherent photoproduction of π⁰-meson from the deuteron is investigated within an approach which includes the reaction amplitudes of the impulse approximation, two-step process with intermediate πN - and ηN-rescattering, and the higher order terms in the multiple scattering series for the intermediate ηNN interaction. This approach is based on realistic separable representations of the driving two-body interaction in the πN , ηN , and NN subsystems. For the elementary γN → πN operator, the unitary isobar model MAID2007 is used which describes well the elemen-tary reaction. The radial deuteron wave function in the initial and final states are  taken from the realistic CD-Bonn potential. Results for unpolarized and polarized cross sections as well as for all possible polarization observables with polarized photons and/or oriented deuterons are predicted and compared with available data, and differences with other theoretical models are analyzed. The contribution of γd → π⁰d to the deuteron spin asymmetry is calculated and its contribution to the Gerasimov-Drell-Hearn (GDH) integral is explicitly evaluated by integration up to a photon lab-energy of 900 MeV. The helicity E-asymmetry is also estimated. This work is motivated by the measurements of the CLAS Collaboration at Jefferson laboratory, where a cusp-like structure in the energy dependence of the differential cross section has been observed near η-production threshold at extremely backward pion angles, and the measurements of the VEPP-3 electron storage ring, where very recent data for the tensor-deuteron spin asymmetries have been measured. The effect of intermediate ηNN three-body interaction is significant in certain spin asymmetries, specially at extreme backward pion angles. It turns out that the inclusion of a full ηNN three-body contribu-tion is quite essential. At forward angles, the contribution from the pure impulse approximation is dominated. The sensitivity of the obtained results to the elementary γN → πN amplitude and the choice of the NN potential model governs the deuteron wave function is investigated and con-siderable dependences are found. Compared to the experimental data from CLAS Collaboration for differential cross section, from YerPhi Collaboration for the linear photon asymmetry, and from VEPP-3 electron storage ring for the tensor target asymmetries, sizeable discrepancies are found.