<s>[INST] Find all triplets of natural numbers $(x,y,z)$ that satisfy this equation: $2x^{2}y^{2}+2y^{2}z^{2}+2x^{2}z^{2}-x^{4} -y^{4}-z^{4}=576$ I've tried 
$(x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(x^{2}-z^{2})^{2}-x^{4}-y^{4}-z^{4}=-576$  

$(x^{2}-y^{2}-z^{2})(x^{2}-y^{2}+z^{2})+(y^{2}-z^{2}-x^{2})(y^{2}-z^{2}+x^{2})+(x^{2}-z^{2}-y^{2})(x^{2}-z^{2}+y^{2})=-576$

I wanted to factor 576 somehow but I cant put the left side of the equation under a common multiplier, please help.
 [/INST] HINT:
$2x^{2}y^{2}+2y^{2}z^{2}+2x^{2}z^{2}-x^{4} -y^{4}-z^{4}$
$=(2xy)^2-(x^2+y^2-z^2)^2$
$=\{2xy+(x^2+y^2-z^2)\}\{2xy-(x^2+y^2-z^2)\}$
$=\{(x+y)^2-z^2\}\{z^2-(x-y)^2\}=\cdots$
Observe that $576$ is even 
and so are the difference & sum of any two of 
$\{x+y+z,x+y-z,z-x+y,z+x-y\}$
So, each of the four multiplicands have to be even.
$$\dfrac{x+y+z}2\cdot\dfrac{x+y-z}2\cdot\dfrac{z+x-y}2\cdot\dfrac{z-x+y}2=\dfrac{576}{2^4}=36$$ 
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