The problem of an elastic stress field in a rotating medium is formulated and solved analytically within the limits of the classical theory of elasticity with a symmetrical stress tensor. This is a rotation elastic field of action at a distance. There are two specific types of elastic waves with a moment in rotating media: solitons and excitons, or rotation waves. The soliton solutions to the wave equation represent waves of global earthquake migration (slow tectonic waves) which are no faster than ∼1 cm/s, i.e., approach the migration velocity of large and great earthquakes (M > 7.5). The exciton solutions correspond to waves of local migration of foreshocks and aftershocks in earthquake sources (fast tectonic waves) and have their maximum velocity comparable to faulting rate and/or to S-wave velocities.