  We consider the Pseudo-Gap (PG) state of high-$T_c$ superconductors in form
of a composite order parameter fluctuating between 2$p_F$-charge ordering and
superconducting (SC) pairing. In the limit of linear dispersion and at the
hotspots, both order parameters are related by a SU(2) symmetry and the eight
hotspot model of Efetov $\textit{et al.}$ [Nat. Phys. $\bf{9}$, 442 (2013)] is
recovered. In the general case however, curvature terms of the dispersion will
break this symmetry and the degeneracy between both states is lifted. Taking
the full momentum dependence of the order parameter into account, we measure
the strength of this SU(2) symmetry breaking over the full Brillouin zone. For
realistic dispersion relations including curvature we find generically that the
SU(2) symmetry breaking is small and robust to the fermiology and that the
symmetric situation is restored in the large paramagnon mass and coupling
limit. Comparing the level splitting for different materials we propose a
scenario that could account for the competition between the PG and the SC
states in the phase diagram of high-$T_c$ superconductors.
