设变换 $\left\{\begin{array}{l}u=x-2 y, \\ v=x+a y\end{array}\right.$ 可把方程 $6 \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial x \partial y}-\frac{\partial^{2} z}{\partial y^{2}}=0$ 简化为 $\frac{\partial^{2} z}{\partial u \partial v}=0$, 求常数 $a$. (这里应假设 $z$ 有二阶连续偏导数. )