Besides $(x, y, z)=(0, 0, 0)$ and $(1, 1, -2)$ (and their permutations) are there any other integer solutions to the equation
$$3(x^{3}+y^{3}+z^{3})+3(x^{2}+y^{2}+z^{2})+(x+y+z)=0 $$ ?
Besides $(x, y, z)=(0, 0, 0)$ and $(1, 1, -2)$ (and their permutations) are there any other integer solutions to the equation
$$3(x^{3}+y^{3}+z^{3})+3(x^{2}+y^{2}+z^{2})+(x+y+z)=0 $$ ?
A simple transformation renders this equivalent to $(3x + 1)^3 + (3y + 1)^3 + (3z + 1)^3 = 3$; any solutions would give solutions to $a^3 + b^3 + c^3 = 3$ (and vice versa, as pointed out by Emil Jeřábek in a comment). According to a recent arxiv article, the only known solutions to the latter are $(1, 1, 1), (4, 4, -5)$ and its permutations, but the existence of other solutions remains open, so this question is also open.
Edited to add: This question seems to have been asked at a very opportune time; Booker and Sutherland have just found another solution: $569936821221962380720^3−569936821113563493509^3−472715493453327032^3 = 3$. That corresponds to $x = 189978940407320793573, y = -189978940371187831170, z = -157571831151109011$ for your question.