$$xy^2=x^4-x+1$$ $$yz^2=y^4-y+1$$ $$zx^2=z^4-z+1$$
My first idea was to prove that all of the three variables are positive but I can't find a way to phrase the whole proof.
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Just as an observation, $x = y = z = 1$ is a solution. – Math Lover Nov 22 '20 at 11:25
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1Also if you subtract one from another, you get $(x-y)(x^3+y^3-1) = y(xy-yz)$ and other two equations of the same form. That at least helps to show $x = y = z = 1$ is a solution. By the way Wolfram Alpha confirms there is only one solution in real which is this. – Math Lover Nov 22 '20 at 11:41