  In this paper, we consider the planar self-affine measures $\mu_{M,D}$
generated by an expanding matrix $M\in M_2(\mathbb{Z})$ and an integer digit
set $ D=\left\{ {\left( {\begin{array}{*{20}{c}} 0\\ 0 \end{array}}
\right),\left( {\begin{array}{*{20}{c}} \alpha_1\\ \alpha_2 \end{array}}
\right),\left( {\begin{array}{*{20}{c}} \beta_1\\ \beta_2 \end{array}} \right)}
\right\} $ with $\alpha_1\beta_2-\alpha_2\beta_1\neq0$. We show that if
$\det(M)\notin 3\mathbb{Z}$, then the mutually orthogonal exponential functions
in $L^2(\mu_{M,D})$ is finite, and the exact maximal cardinality is given.
