<s>[INST] How to solve this symmetric system of equations? How many solutions are there to this equation?
$$\begin{align*}
x^2-y^2&=z\\
y^2-z^2&=x\\
z^2-x^2&=y
\end{align*}$$
 [/INST] We have
\begin{align}
x^2 - y^2 & = z\\
y^2 - z^2 & = x\\
z^2 - x^2 & = y
\end{align}
Adding the first two equations, we get that
$$x^2-z^2 = z+x \implies z=-x \text{ or }x-z=1$$
Similarly, we get that
$$x=-y \text{ or }y-x=1$$
$$y=-z \text{ or }z-y=1$$
Hence, there are $8$ possible choices of getting $3$ equations. But the choice
$$x-z=1 \text{ and }y-x=1 \text{ and }z-y=1$$ gives no solution. The rest of the seen choices give the following solutions
$$(0,0,0); (-1,0,1); (0,1,-1); (1,-1,0)$$
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