On seventh powers $x_1^7+x_2^7+\dots+x_n^7 = 2$? We have,
$$(-6m^3 + 1)^3 + (6m^3 + 1)^3 + (-6m^2)^3 = 2$$
$$(-8m^5 + 1)^5 + (8m^5 + 1)^5 + (-8m^6 + 2m)^5 + (-8m^6 - 2m)^5 + 2(8m^6)^5=2$$
The first identity has been long known, while the second is by Ajai Choudhry. Anybody knows if there is anything similar for 7th powers?

The best I can do is in terms of the radical $\sqrt{3}$:
$$2=(9 m^7 + 1)^7 + (-9 m^7 + 1)^7 + (\sqrt{3} m - 9 m^8)^7 + (-\sqrt{3} m - 
    9 m^8)^7 + 2 (9 m^8)^7$$
