of order 1 (Theorem \[Theorem:K1\]), thereby extending the Kohn-Müller result to the present vectorial model. The presence of a second rank-one connection becomes, however, important in the case of long and thin domains, in which microstructure is generated along vertical oscillations instead of horizontal ones, as we discuss after Theorem \[Theorem:K1\] and in Section \[secubtheo1\] below.

The second situation we consider, denoted case $K_2$, is $$\label{assumption:K2}
A_2={\ensuremath{\begin{pmatrix}1&0 \\ 0&1-\alpha \end{pmatrix}}}, \quad B_2={\ensuremath{\begin{pmatrix}1&0 \\ 0&1+\alpha \end{pmatrix}}}, \quad K_2=\operatorname{SO}(2)A_2\cup\operatorname{SO}(2)B_2$$ for some $\alpha\in(0,1)$. For $\alpha=1$, the matrix $A_2$ becomes singular; treating this unphysical case would generate some complications in the proofs, for clarity we restrict in the relevant parts to $\alpha<1/2$. Clearly, any other condition $\alpha<\alpha_*$, for any fixed $\alpha_*<1$, would lead to the same results. Physically relevant values of the spontaneous strain are typically of a few percent, up to about $10\%$ [@B9_Bhatta; @OtsukaWayman1999], and therefore well within the range $(0,1/2)$.

The matrices in (\[assumption:K2\]) have a different determinant and are representative of phase transformations with one rank-one connection, in the sense that $\det(A_2-QB_2)=0$ has a unique solution $Q\in 
\operatorname{SO}(2)$. In this case, the invariance under rotations and the vectorial nature of the deformation can interact to reduce the energy cost of 