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The third planform is fixed by a diagonal reflection and changes sign under a quarter-turn. Extend the preceding group representation byThe other diagonal reflection changes only the sign of $B$. Consequently the first two components of an equivariant dynamical system have factors $A$ and $B$, whereas its third component is even in each of $A,B$. The quarter-turn exchanges $A^2$ and $B^2$ while negating $C$. Its admissible quadratic coupling is therefore $A^2-B^2$ in the third equation and opposite $AC,BC$ couplings in the first two. At cubic order the only new terms are $A{C}^2,B{C}^2,C^3,C(A^2+B^2)$. The linear action on the two-dimensional irreducible mode has one coefficient ${\mu }_1$, while the distinct scalar representation has its own coefficient ${\mu }_2$. A small splitting of the two marginal linear modes supplies these two unfolding parameters. With real coefficients the resulting amplitude equations areThese equations follow by listing all equivariant monomials through degree three; the displayed minus signs are coefficient conventions, not consequences of symmetry.

In the specified reduced system put $A=B=s$, $C=0$. A nonzero branch exists for ${\mu }_1>0$, with $q=s^2={\mu }_1/(1+\lambda )$. Write perturbations as $u=(\delta A+\delta B)/2$, $v=(\delta A-\delta B)/2$, and $c=\delta C$. Linearization givesFor the square-pattern interaction with a sign-changing scalar mode, the remaining eigenvalues have sum $\tau =p+{\mu }_2$ and product $D=p{\mu }_2+4{\alpha }_1{\gamma }_{1}q=p({\mu }_2+K)$, where $K=2{\alpha }_1{\gamma }_1/(\lambda -1)>0$. A real two-dimensional stability matrix has both eigenvalues in the left half-plane exactly when its trace is negative and its determinant is positive. ThereforeIf ${\mu }_2<-K$, the block has negative determinant and one growing direction, explaining the instability at sufficiently small ${\mu }_2$. At the lower endpoint ${\mu }_2=-K$, one eigenvalue is zero and the other is $p-K<0$. The symmetry $(A,B,C)\mapsto (B,A,-C)$ fixes the diagonal state and negates its critical $(v,c)$ direction. Thus a generic nonlinear unfolding has a pitchfork bifurcation to unequal-amplitude states with nonzero $C$.

At the upper endpoint ${\mu }_2=-p$, the trace vanishes andThe symmetric eigenvalue remains negative. The critical conjugate pair crosses the imaginary axis with derivative $d\mathrm{Re}\sigma /d{\mu }_2=1/2$. If its cubic Hopf normal form coefficient is nonzero, this is a Hopf bifurcation and nearby periodic solutions exist on one side. Their stability and the side of existence depend on that coefficient; the linear calculation does not determine them. It does establish the generic possibility of oscillating square patterns. The equality $p=K$ is a double-zero degeneracy, outside this nondegenerate Hopf bifurcation argument.