  In this work, we employ the $SO(2)$-rotations of a two-component, one-, two-
and three-dimensional nonlinear Schr\"{o}dinger system at and near the Manakov
limit, to construct vector solitons and vortex structures. This way, stable
stationary dark-bright solitons and their higher-dimensional siblings are
transformed into robust oscillatory dark-dark solitons (and generalizations
thereof), with and without a harmonic confinement. By analogy to the
one-dimensional case, vector higher-dimensional structures take the form of
vortex-vortex states in two dimensions and, e.g., vortex ring-vortex ring ones
in three dimensions. We consider the effects of unequal (self- and cross-)
interaction strengths, where the $SO(2)$ symmetry is only approximately
satisfied, showing the dark-dark soliton oscillation is generally robust.
Similar features are found in higher dimensions too, although our case examples
suggest that phenomena such as phase separation may contribute to the
associated dynamics. These results, in connection with the experimental
realization of one-dimensional variants of such states in optics and
Bose-Einstein condensates (BECs), suggest the potential observation of the
higher-dimensional bound states proposed herein.
