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\begin{aligned} &\mathscr{L}_{x} z=\partial_{x}^{2} z+\partial_{x} \partial_{y} z+f(x) \partial_{x} z \\ &\mathscr{L}_{y} z=\partial_{y}^{2} z+\partial_{x} \partial_{y} z+f(y) \partial_{y} z \end{aligned}