  We construct families of ground state representations of the U(1)-current net
and of the Virasoro nets Vir_c with central charge c >= 1. We show that these
representations are not covariant with respect to the original dilations, and
those on the U(1)-current net are not solitonic.
  Furthermore, by going to the dual net with respect to the ground state
representations of Vir_c, we obtain a possibly new family of M\"obius covariant
nets on S^1.
