【解析】由多元复合函数求导法则, 得
\begin{aligned} &\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial x}+\frac{\partial z}{\partial v} \frac{\partial v}{\partial x}=\frac{\partial z}{\partial u}+\frac{\partial z}{\partial v} \\ &\frac{\partial z}{\partial y}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial y}+\frac{\partial z}{\partial v} \frac{\partial v}{\partial y}=-2 \frac{\partial z}{\partial u}+a \frac{\partial z}{\partial v} \end{aligned}

\begin{aligned} \frac{\partial^{2} z}{\partial x^{2}} &=\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial u}\right)+\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial v}\right)=\frac{\partial^{2} z}{\partial u^{2}} \cdot \frac{\partial u}{\partial x}+\frac{\partial^{2} z}{\partial u \partial v} \cdot \frac{\partial v}{\partial x}+\frac{\partial^{2} z}{\partial v^{2}} \cdot \frac{\partial v}{\partial x}+\frac{\partial^{2} z}{\partial v \partial u} \frac{\partial u}{\partial x} \\ &=\frac{\partial^{2} z}{\partial u^{2}}+2 \frac{\partial^{2} z}{\partial u \partial v}+\frac{\partial^{2} z}{\partial v^{2}} \\ \frac{\partial^{2} z}{\partial x \partial y} &=\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial u}\right)+\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial v}\right)=\frac{\partial^{2} z}{\partial u^{2}} \cdot \frac{\partial u}{\partial y}+\frac{\partial^{2} z}{\partial u \partial v} \cdot \frac{\partial v}{\partial y}+\frac{\partial^{2} z}{\partial v^{2}} \cdot \frac{\partial v}{\partial y}+\frac{\partial^{2} z}{\partial v \partial u} \frac{\partial u}{\partial y} \\ &=-2 \frac{\partial^{2} z}{\partial u^{2}}+(a-2) \frac{\partial^{2} z}{\partial u \partial v}+a \frac{\partial^{2} z}{\partial v^{2}}, \\ \frac{\partial^{2} z}{\partial y^{2}} &=-2 \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial u}\right)+a \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial v}\right) \\ &=-2\left(\frac{\partial^{2} z}{\partial u^{2}} \cdot \frac{\partial u}{\partial y}+\frac{\partial^{2} z}{\partial u \partial v} \cdot \frac{\partial v}{\partial y}\right)+a\left(\frac{\partial^{2} z}{\partial v^{2}} \cdot \frac{\partial v}{\partial y}+\frac{\partial^{2} z}{\partial v \partial u} \cdot \frac{\partial u}{\partial y}\right) \\ &=4 \frac{\partial^{2} z}{\partial u^{2}}-4 a \frac{\partial^{2} z}{\partial u \partial v}+a^{2} \frac{\partial^{2} z}{\partial v^{2}} \end{aligned}

$$6 \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial x \partial y}-\frac{\partial^{2} z}{\partial y^{2}}=(10+5 a) \frac{\partial^{2} z}{\partial u \partial v}+\left(6+a-a^{2}\right) \frac{\partial^{2} z}{\partial v^{2}}=0$$

$a=-2$ 时, $10+5 a=0$, 故舍去, $a=3$ 时, $10+5 a \neq 0$, 因此仅当 $a=3$ 时化简为 $\frac{\partial^{2} z}{\partial u \partial v}=0$.
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