Velocity addition and a closed time cycle in Lorentz-noninvariant theories

In theories whose Lorentz invariance is violated by the presence of an external tensor of any rank, we show that a signal velocity, understood as the group velocity of a wave, is added to the velocity of the reference frame according to the standard relativistic rule for adding velocities. In the case where we have a superluminal signal, this observation allows creating a closed time cycle and thus coming to a conclusion about a causality violation even in the absence of relativistic invariance. We also reveal an optical anisotropy of a moving medium that is isotropic at rest.