Exact Results on Equations of Motion in Vacuum String Field Theory
  We prove some algebraic relations on the translationally invariant solutions and the lump solutions in vacuum string field theory. We show that up to the subtlety at the midpoint the definition of the half-string projectors of the known sliver solution can be generalized to other solutions. We also find that we can embed the translationally invariant solution into the matrix equation of motion with the zero mode. 
Hiroyuki Hata and Sanefumi Moriyama