Suppose that $N$ is coprime with $6$ and that $-3$ is a square mod $N$, say $\rho^2=-3\pmod{N}$.  Put $\omega=(-1+\rho)/2$ so $\omega^2=\omega^{-1}=-1-\omega=(-1-\rho)/2$ in $\mathbb{Z}/N$.  Put 
\begin{align*}
 X &= (U+V+W)/3 \\
 Y &= (U+\omega V+\omega^2W)/3 \\
 Z &= (U+\omega^2V+\omega W)/3
\end{align*}
Then your equation becomes $UVW=1\pmod{N}$, so you can take $U$ and $V$ to be arbitrary numbers that are coprime to $N$, and $W=1/(UV)\pmod{N}$.