Classification of $\lambda$-homomorphic braces on $\mathbb{Z}^2$

If $A=(A,\oplus,\odot)$ is a $\lambda$-homomorphic brace with $(A,\oplus)=\mathbb{Z}^2$, then the operations in this brace are given by formulas \begin{align*}\begin{pmatrix}a_1\\a_2\end{pmatrix}\oplus\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1+b_1\\a_2+b_2\end{pmatrix},&&\begin{pmatrix}a_1\\a_2\end{pmatrix}\odot\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1\\a_2\end{pmatrix}+\varphi^{a_1}\psi^{a_2}\begin{pmatrix}b_1\\b_2\end{pmatrix}, \end{align*} where $\varphi,\psi\in{\rm GL}_2(\mathbb{Z})$ are cpecific matrices which depend on $A$. Not every pair $(\varphi,\psi)$ lead to a brace. In the present paper we find all possible pairs $(\varphi,\psi)$ of matrices from ${\rm GL}_2(\mathbb{Z})$ which lead to $\lambda$-homomorphic braces with $(A,\oplus)=\mathbb{Z}^2$. The obtained result gives the full classification of $\lambda$-homomorphic braces on $\mathbb{Z}^2$ which was started by Bardakov, Neshchadim and Yadav in [J. Pure App. Algebra, V. 226, N. 6, 2022, 106961].