<s>[INST] Is it possible to generalize this equation more? This is one of my discoveries, and my question is: can it be more general?
Let $x, y, z$ be three arbitrary complex numbers and set
$x_1=3 x^2+y^2$,
$x_2=1-z$,
$x_3=1+z$,
$x_4=3 x^2+y^2+4 x y z$,
$x_5=3 x^2+y^2-4 x y z$,
$x_6=3 x^2+y^2+z (3 x^2+2 x y-y^2)$,
$x_7=3 x^2+y^2-z (3 x^2+2 x y-y^2)$,
$x_8=3 x^2+y^2+z (3 x^2-2 x y-y^2)$,
$x_9=3 x^2+y^2-z (3 x^2-2 x y-y^2)$.
Then for $k=0,1,2,3,4,5$,
$2(x_1^k+x_1^kx_2^k+x_1^kx_3^k)=x_4^k+x_5^k+x_6^k+x_7^k+x_8^k+x_9^k$.
In case $x, y, z$ are integers, we get a Diophantine equation.
For example,
$2(7^1+21^1+(-7)^1)=(-9)^1+23^1+1^1+13^1+17^1+(-3)^1$,
$2(7^2+21^2+(-7)^2)=(-9)^2+23^2+1^2+13^2+17^2+(-3)^2$,
$2(7^3+21^3+(-7)^3)=(-9)^3+23^3+1^3+13^3+17^3+(-3)^3$,
$2(7^4+21^4+(-7)^4)=(-9)^4+23^4+1^4+13^4+17^4+(-3)^4$,
$2(7^5+21^5+(-7)^5)=(-9)^5+23^5+1^5+13^5+17^5+(-3)^5$.
My discovery published at: https://demonstrations.wolfram.com/author.html?author=Minh%20Trinh%20Xuan
 [/INST] This is very nice. It relies to the Tarry-Escott problem, on which there is a vast literature going back more than 100 years. I do not know how  much you know of this. I suggest you consult Dickson, the History of the Theory of Numbers.
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