= Solution
The third planform is fixed by a diagonal <reflection> and changes sign under a quarter-turn. Extend the preceding <group representation> by
$$
r(A,B,C)=(B,-A,-C),\qquad s(A,B,C)=(-A,B,C).
$$
The 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 $AC^2,BC^2,C^3,C(A^2+B^2)$. 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> are
$$
\begin{aligned}
A_T&=\mu_1A-\alpha_0A^3-\lambda AB^2-\alpha_1AC-\alpha_2AC^2,\\
B_T&=\mu_1B-\alpha_0B^3-\lambda A^2B+\alpha_1BC-\alpha_2BC^2,\\
C_T&=\mu_2C-\gamma_0C^3+\gamma_1(A^2-B^2)-\gamma_2C(A^2+B^2).
\end{aligned}
$$
These 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> gives
$$
u_T=-2\mu_1u,\qquad
\begin{pmatrix}v\\c\end{pmatrix}_T=
\underbrace{\begin{pmatrix}p&-\alpha_1s\\4\gamma_1s&\mu_2\end{pmatrix}}_{M}
\begin{pmatrix}v\\c\end{pmatrix},\qquad p=2(\lambda-1)q>0.
$$
For the <square-pattern interaction with a sign-changing scalar mode>, the remaining <eigenvalues> have sum $\tau=p+\mu_2$ and product $D=p\mu_2+4\alpha_1\gamma_1q=p(\mu_2+K)$, 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. Therefore
$$
\boxed{-K<\mu_2<-p,\qquad 0<\mu_1<\frac{(1+\lambda)\alpha_1\gamma_1}{(\lambda-1)^2}.}
$$
If $\mu_2<-K$, 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 and
$$
\boxed{\omega_H^2=\det M=p(K-p)>0.}
$$
The symmetric <eigenvalue> remains negative. The critical conjugate pair crosses the imaginary axis with <derivative> $d\operatorname{Re}\sigma/d\mu_2=1/2$. 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.
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