  This paper continues the investigation of the formation of naked
singularities in the collapse of collisionless matter initiated in [RV]. There
the existence of certain classes of non-smooth solutions of the Einstein-Vlasov
system was proved. Those solutions are self-similar and hence not
asymptotically flat. To obtain solutions which are more physically relevant it
makes sense to attempt to cut off these solutions in a suitable way so as to
make them asymptotically flat. This task, which turns out to be technically
challenging, will be carried out in this paper.
  [RV] A. D. Rendall and J. J. L. Vel\'{a}zquez, A class of dust-like
self-similar solutions of the massless Einstein-Vlasov system. Annales Henri
Poincare 12, 919-964, (2011).
