Find $ k \in \mathbb{N}$ such that $x^3+y^3+z^3=kx^2y^2z^2$ have positive integer root

Question

Find $k \in \mathbb{N}$ such that $x^3+y^3+z^3=kx^2y^2z^2$ have positive integer roots

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I know a similar problem $x^3 + y^3 + z^3 = nxyz$ but I still can't solve my problem

Answer 1

Do you want all $k$, or will you settle for a couple of them?

$k=1$ works with $\{{x,y,z\}}=\{{1,2,3\}}$.

$k=3$ works for $x=y=z=1$.

Getting all solutions, that may be asking for too much. These higher degree Diophantine equations can be tough.