Let $x,y$ be odd integers and let $z$ be an integer. The question is to find all solutions to the equation, $$x^3+y^3+4z^3=0$$ Of course we have the trivial solution $(x,-x,0)$. Are there any others?
By considering the equation modulo $4$ we see that wlog $x=4k+1$ and $y=4m+3$. Numerical experimentation shows that there are no solutions with $|k|, |m| < 10000$. In fact since the cubic residues modulo $9$ are $0,\pm 1$ then we see that $z$ is a multiple of $9$ and moreover, dividing by $9^3$ if necessary, we see that $x^3$ and $y^3$ must have opposite residue modulo $9$. The same holds by considering the equation modulo $7$. I'm also aware of Broughan's theorem, but it's not clear to me whether it helps.