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Integral points on "complex exponential surface" in $\mathbb{C}^3$

I encountered the following object in $\mathbb{C}^3$ defined for $m\in\mathbb{N}$ by $$A_m:=\lbrace (z_1,z_2,z_3)\in\mathbb{C}^3|(2^{2z_3}m-1)2^{2z_1+z_2+1}+3^{z_2-1}(2^{2z_1}-2^2-3^{z_3+1}m)=0\rbrace$...
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