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 of with respect to the ground state representations of Vir_c, one obtains possibly new family of Möbius covariant nets on S¹.