Solve in $\mathbb{Z}^4$ $$x^3+y^3+z^3=txyz$$ (Ion Ionescu, 1931)
This is the problem.
What I tried to do:
$t=3+k\Rightarrow$ $$(x+y+z)(x^2+y^2+z^2-xy-xz-yz)=kxyz$$
For $k = 0$ we have infinite solutions. For $k\ne0$ I have no idea.
Please help, I am confused.
A very nice and elementary solution is given in the article
Solutions of $x^3+y^3+z^3=n\cdot xyz$
by Erik Dofs. It contains references and tables with special solutions for some $n$. For example, for $n=73$ we have $$ 89200900157319^3+ 2848691279889518^3+ 1391526622949983^3=n\cdot xyz $$
