  We show that the fixed point subnet of a strongly additive conformal net
under the action of a compact group is strongly additive. Using the idea of the
proof we define the notion of strong additivity for a pair of conformal nets
and we show that a key result about the induction of the pair which we proved
previously under the finite index assumption can be generalized to strongly
additive pairs of conformal nets. These results are applied to classify
conformal nets of central charge $c=1$ which are not necessarily rational and
satisfy a spectrum condition.
