$a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ac)$, so $abc= 0$. Say WLOG $a=0$, so $b+c = 0$ and $b^2+c^2 = 1$. We find $(a,b,c) = (0,\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}})$ and its permutations.