Lines of fixed points and physically irreducible representations

We will prove that if the order parameter belongs to the representation $D$ which is the direct sum of two complex conjugate irreducible representations, $D=\Delta \oplus {\Delta }^{*}$, and if $\Delta$ has at least one quartic invariant, then the symmetry SO(2) appears in the parameter space. If, in addition, $\Delta$ and ${\Delta }^{*}$ are quasiequivalent then higher symmetry than SO(2) will appear in the parameter space. Since SO(2) is a continuous group, every fixed point of the renormalization-group transformations will be either invariant under SO(2) or part of a fixed line generated by SO(2) transformations on the fixed point.