  We show that any solitonic representation of a conformal (diffeomorphism
covariant) net on S^1 has positive energy and construct an uncountable family
of mutually inequivalent solitonic representations of any conformal net, using
nonsmooth diffeomorphisms. On the loop group nets, we show that these
representations induce representations of the subgroup of loops compactly
supported in S^1 \ {-1} which do not extend to the whole loop group.
  In the case of the U(1)-current net, we extend the diffeomorphism covariance
to the Sobolev diffeomorphisms D^s(S^1), s > 2, and show that the
positive-energy vacuum representations of Diff_+(S^1) with integer central
charges extend to D^s(S^1). The solitonic representations constructed above for
the U(1)-current net and for Virasoro nets with integral central charge are
continuously covariant with respect to the stabilizer subgroup of Diff_+(S^1)
of -1 of the circle.
