I apologize as I just realized the reply to this POTW #413 has gone amiss.
I just checked and no one answered to this POTW. Nevertheless, you can refer to the suggested solution by other below:
We have $xyz=x^2+y^2+z^2>Y^2+z^2\ge 2yz$. Hence $x>2$ and $x^3>x$. Similarly, $y^3>y$ and $z^3>z$. Adding them up gives $x^3+y^3+z^3>x+y+z$ and this contradicts to what is given and hence, there are no solutions to the system.