A generalization of the slow-motion theory of nuclear spin relaxation in paramagnetic systems (S=1) is developed. The new model takes into account the effects of rhombic symmetry in the static zero-field splitting tensor. We also allow the principal axis system of the static zero-field splitting tensor to deviate from the molecule-fixed frame of the dipole-dipole tensor between the nuclear and electron spins. These symmetry-breaking properties have profound effects on the nuclear spin-lattice relaxation rate for some cases. Specifically, the relaxivity is reduced substantially at low magnetic field. Nuclear magnetic relaxation dispersion profiles for a large number of cases are discussed, ranging from slightly asymmetric [low static zero-field splitting (ZFS)] weakly deformable (low transient ZFS) to asymmetric (large static ZFS) highly deformable (large transient ZFS) transition-metal complexes. The dynamical regimes covered for the electron spin range from within the Redfield limit into the slow-motion region. One of the main objectives of this investigation is to provide a standard set of essentially exact calculations using the general slow-motion theory, against which simplified models may be tested.