A shadow system  appears as a limit of  a reaction-diffusion system in
which some  components have infinite diffusivity.   We investigate the
spatial structure of  its stable solutions.  It is  known that, unlike
scalar  reaction-diffusion  equations, some  shadow  systems may  have
stable  nonconstant (monotone) solutions.   On the  other hand,  it is
also  known   that  in  autonomous  shadow   systems  any  nonconstant
non-monotone  stationary  solution is  necessarily  unstable. In  this
paper, it is  shown in a general setting that  any stable bounded (not
necessarily  stationary)  solution  is asymptotically  homogeneous  or
eventually monotone in $x$.