sol = Sort@
  DeleteDuplicates@
   Cases[Table[
     Reduce[Floor[2 Cos[x]] == i && 0 < x < 2 Pi, x], {i, -2, 2}],
   _Times, Infinity]

Plot[Floor[2 Cos[x]], {x, 0, 2 Pi}, Mesh -> {Range[-2, 2]},
 MeshFunctions -> {#2 &}, MeshStyle -> Directive[Red, PointSize[.02]],
  Ticks -> {ticks, Automatic}, Filling -> Axis]