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We state the conditions under which the general spin-s quantum XYZ ferromagnet (H−) and antiferromagnet (H+) with an external magnetic field along one axis, specified by the Hamiltonian H±=± 𝒥Nl=1 (JxSxlSxl+1+JySylS l+1y+JzSzlSzl+1)-h 𝒥Nl=1Szl exhibits a fully ordered ground state described by a wave function which is a direct product of single-site wave functions. We present a detailed analysis of the implications for the zero-temperature dynamical properties of this model. In particular, we derive a rigorous relation between the three dynamic structure factors Sμμ(q,ω), μ=x,y,z at T=0. For the special case of the s=(1/2) anisotropic XY model (Jz=0), these relations are used to determine the dynamic structure factors Sxx(q,ω) and Syy(q,ω) at T=0 and h=(JxJy)1/2 in terms of the known dynamic structure factor Szz(q,ω).