Suppose we have an indefinite equation about $$$x,y,z$$$.

If $$$(x,y,z)=(2,2,2)$$$ satisfies the equation, is there other such $$$(x,y,z)$$$ groups with $$$x,y,z \geq 2$$$ satisfy the equation?

Notice, that unlike Fermat's last theorem, $$$x,y,z$$$ does NOT have to be pairwise equal.

I think that the answer is NO, but I am unable to prove it or construct such groups. Please help me :D