  In this work we present a class of traveling solitons in Lorentz-violating
systems. In the case of Lorentz violating scenarios, it is usual to construct
static solitonic configurations. Here it is shown that it is possible to
construct some traveling solitons which, as it should be expected, can not be
mapped into a static configuration by means of Lorentz boosts due to its
explicit breaking. Furthermore, in the model studied, a complete set of
solutions is obtained. The solutions present a critical behavior controlled by
the choose of an arbitrary integration constant.
