Quantum spin-chains subjected to external oscillating fields exhibit a wide range of dynamical behavior depending on the strength and frequency of the field. We investigate here the particular case of a Heisenberg model with a single spin flip; this maps onto a tight-binding model with an oscillating field, which, in turn, is equivalent to a driven-Harper model with sinusoidal time-driving. The Harper models can be mapped to image classical phasespaces. We find that peculiar structures called "shearless tori" feature very prominently in the classical phase space of the driven-Harper model. We show that quantum-spin wave packets initially located on a "shearless torus" can travel for a quite long time without broadening. The behavior has potential applications as it may provide a mechanism for non-dispersive transfer of quantum information.