http://mathworld.wolfram.com/DiophantineEquation4thPowers.html (equation 144) gives Ramanujan's
$$a^4(b-c)^4+ b^4(c-a)^4+ c^4(a-b)^4= 2(ab+bc+ca)^4$$
where $a+b+c=0$.
And http://mathworld.wolfram.com/FerrarisIdentity.html gives the Ferrari identity
$$(a^2+2ac-2bc-b^2)^4+ (b^2-2ba-2ca-c^2)^4+ (c^2+2cb+2ab-a^2)^4 = 2(a^2+b^2+c^2-ab+bc+ca)^4$$