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Motivated by the recently synthesized insulating nickelate Ni$_2$Mo$_3$O$_8$,which has been reported to have an unusual non-collinear magnetic order ofNi$^{2+}$ $S=1$ moments with a nontrivial angle between adjacent spins, weconstruct an effective spin-1 model on the honeycomb lattice, with the exchangeparameters determined with the help of first principles electronic structurecalculations. The resulting bilinear-biquadratic model, supplemented with therealistic crystal-field induced anisotropy, favors the collinear N\'eel state.We find that the crucial key to explaining the observed noncollinear spinstructure is the inclusion of the Dzyaloshinskii--Moriya (DM) interactionbetween the neighboring spins. By performing the variational mean-field andlinear spin-wave theory (LSWT) calculations, we determine that a realisticvalue of the DM interaction $D\approx 2.78$ meV is sufficient to quantitativelyexplain the observed angle between the neighboring spins. We furthermorecompute the spectrum of magnetic excitations within the LSWT and random-phaseapproximation (RPA) which should be compared to future inelastic neutronmeasurements.

Noncollinear Antiferromagnetic Order and Effect of Spin-Orbit Coupling in Spin-1 Honeycomb Lattice