Fourier bases of a class of planar self-affine measures

Let ${\mu }_{M,D}$ be the planar self-affine measure generated by an expansive integer matrix $M\in M_2(\mathbb{Z})$ and a non-collinear integer digit set $D=\{\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right),\left(\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right),\left(\begin{array}{c}-{\alpha }_1-{\beta }_1\\ -{\alpha }_2-{\beta }_2\end{array}\right)\}$. In this paper, we show that ${\mu }_{M,D}$ is a spectral measure if and only if there exists a matrix $Q\in M_2(\mathbb{R})$ such that $(\tilde{M},\tilde{D})$ is admissible, where $\tilde{M}=QM{Q}^{-1}$ and $\tilde{D}=QD$. In particular, when ${\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1\mathrm{\notin }2\mathbb{Z}$, ${\mu }_{M,D}$ is a spectral measure if and only if $M\in M_2(2\mathbb{Z})$.