  The $Z_2$ bosonic orbifold models with compactification radius $R^2=1/2k$ are
examined in the presence of boundaries.
  Demanding the extended algebra characters to have definite conformal
dimension and to consist of an integer sum of Virasoro characters, we arrive at
the right splitting of the partition function. This is used to derive a free
field representation of a complete, consistent set of boundary states, without
resorting to a basis of the extended algebra Ishibashi states. Finally the
modules of the extended symmetry algebra that correspond to the finitely many
characters are identified inside the direct sum of Fock modules that constitute
the space of states of the theory.
