  A great variety of nonlinear dissipative systems are known to host structures
having a correlation range much shorter than the size of the system. The
dynamics of these Localized Structures (LSs) have been investigated so far in
situations featuring parity symmetry. In this letter we extend this analysis to
systems lacking of this property. We show that the LS drifting speed in a
parameter varying landscape is not simply proportional to the parameter
gradient, as found in parity preserving situations. The symmetry breaking
implies a new contribution to the velocity field which is a function of the
parameter value, thus leading to a new paradigm for LSs manipulation. We
illustrate this general concept by studying the trajectories of the LSs found
in a passively mode-locked laser operated in the localization regime. Moreover,
the lack of parity affects significantly LSs interactions which are governed by
asymmetrical repulsive forces.
