I have the following system of ODEs:
\begin{cases} \frac{du_{i}}{dt}=F_{i}\left(\boldsymbol{u},\boldsymbol{v}\right)+A, & i=1,\ldots M\\ \\ \frac{dv_{j}}{dt}=G_{j}\left(\boldsymbol{u},\boldsymbol{v}\right), & j=1,\ldots N \end{cases}
where I have defined $\boldsymbol{u}\overset{\mathrm{def}}{=} \left(u_{1},\ldots,u_{M}\right)$ and $\boldsymbol{v}\overset{\mathrm{def}}{=} \left(v_{1},\ldots,v_{N}\right)$, while $A$ is a parameter that I want to vary continuously in order to generate bifurcations in the dynamics of the system. Moreover the $M$ equations in $u_{i}$ are symmetric under exchange of the index $i$, while the $N$ equations in $v_{j}$ are symmetric under exchange of the index $j$. In other terms, they are invariant under the action of the group $S_{M}\times S_{N}$, where $S_{M}$ represents the permutation group on $M$ items. This invariance provides the following constraint on the functions $F_{i}\left(\boldsymbol{u},\boldsymbol{v}\right)$ and $G_{j}\left(\boldsymbol{u},\boldsymbol{v}\right)$:
\begin{array}{c} F_{p}\left(\left(P_{pq}^{M},P_{rs}^{N}\right)\cdot\left(\boldsymbol{u},\boldsymbol{v}\right)\right)=F_{q}\left(\boldsymbol{u},\boldsymbol{v}\right)\;\forall r,s\\ \\ G_{r}\left(\left(P_{pq}^{M},P_{rs}^{N}\right)\cdot\left(\boldsymbol{u},\boldsymbol{v}\right)\right)=G_{s}\left(\boldsymbol{u},\boldsymbol{v}\right)\;\forall p,q \end{array}
where $P_{pq}^{M}$ is the permutation between the $p$th and the $q$th item of the population with $M$ elements. In other terms $\left(P_{pq}^{M},P_{rs}^{N}\right)\in S_{M}\times S_{N}$ and $\left(P_{pq}^{M},P_{rs}^{N}\right)\cdot\left(\boldsymbol{u},\boldsymbol{v}\right)=\left(P_{pq}^{M}\boldsymbol{u},P_{rs}^{N}\boldsymbol{v}\right)$, with $P_{pq}^{M}\boldsymbol{u}=P_{pq}^{M}\left(u_{1},\ldots,u_{p},\ldots,u_{q},\ldots,u_{M}\right)=\left(u_{1},\ldots,u_{q},\ldots,u_{p},\ldots,u_{M}\right)$. In this way, since we have also $\left(P_{pq}^{M},P_{rs}^{N}\right)\left(\frac{du_{p}}{dt},\frac{dv_{r}}{dt}\right)=\left(\frac{du_{q}}{dt},\frac{dv_{s}}{dt}\right) $, the system is invariant under the action of the group $S_{M}\times S_{N}$.
Now let's suppose that for a given value of the parameter $A$ we know the stationary solution (i.e. a stable principal branch) of the ODE system, that we call $\left(\overline{\boldsymbol{u}}^{\mathrm{pri}},\overline{\boldsymbol{v}}^{\mathrm{pri}}\right) $. Now, if I vary $A$, I get a branching point bifurcation, so that the solution on the principal branch becomes unstable, and two new solutions (secondary and tertiary branches) emerge from it. We call these new solutions $\left(\overline{\boldsymbol{u}}^{\mathrm{sec}},\overline{\boldsymbol{v}}^{\mathrm{sec}}\right)$ and $\left(\overline{\boldsymbol{u}}^{\mathrm{ter}},\overline{\boldsymbol{v}}^{\mathrm{ter}}\right)$ respectively.
The key point is: we get $P_{pq}^{N}\overline{\boldsymbol{u}}^{\mathrm{sec}}=\overline{\boldsymbol{u}}^{\mathrm{sec}}=\overline{\boldsymbol{u}}^{\mathrm{ter}}$ for any pair $p,q$ (identical coordinates in both $\overline{\boldsymbol{u}}^{\mathrm{sec}}$ and $\overline{\boldsymbol{u}}^{\mathrm{ter}}$), but $\overline{\boldsymbol{v}}^{\mathrm{sec}}\neq\overline{\boldsymbol{v}}^{\mathrm{ter}}$ with $P_{rs}^{N}\overline{\boldsymbol{v}}^{\mathrm{sec}}=\overline{\boldsymbol{v}}^{\mathrm{ter}}$ for some pair $r,s$ (in other terms, two coordinates in the vectors $\overline{\boldsymbol{v}}^{\mathrm{sec}}$, $\overline{\boldsymbol{v}}^{\mathrm{ter}}$ are exchanged). This is clearly a symmetry breaking (in the context of group theory), but is it spontaneous or not?
Thanks in advance for your help!