How to solve system equation $x^3+y^3+z^3=x+y+z, x^2+y^2+z^2=xyz$ for $x,y,z\in\mathbb{R}$? How to solve system equation $\left\{\begin{matrix}x^3+y^3+z^3=x+y+z&\\x^2+y^2+z^2=xyz&\end{matrix}\right.$ , $x,y,z\in\mathbb{R}$ ?

We can show that there is no solution for $x,y,z>0$. Firstly we have:
$$
x^{2}+y^{2}+z^{2}=xyz \implies xyz\geq 3(xyz)^{\frac 23}\\
\implies xyz\geq 27.
$$
On the other hand we have:
$$
x^{2}+y^{2}+z^{2}=xyz\geq \frac 13(x+y+z)^{2}=\frac 13(x^3+y^3+z^3)^{2}\geq \frac 13(3xyz)^{2}=3(xyz)^2\\
xyz\leq \frac 13
$$
So we have $xyz\leq \frac 13$ and $xyz\geq 27$ which is a contradiction and hence there is no solution for positive $x,y,z$. 
