Some interesting systems of equations 1, Solve the system of equations:$\left\{\begin{matrix}
x^3+y^3+2z^3=19x-11y-5z+1\\ x^3+(y^2+1)x=(x^2+y^2)z+z
\\ \sqrt{2+x^2+y^2-2yz}=y^2+z^2-2xy+\sqrt{2}
\end{matrix}\right.$
2,Solve the system of equations:$\left\{\begin{matrix}
x-3z^2x-3z+z^3=0\\ y-3x^2y-3x+x^3=0
\\ z-3y^2z-3y+y^3=0
\end{matrix}\right.$.
The second exercise is in symmetrical performance. But it has three one: x,y,z. How to change it to easier performance? It is one part on my test but i can't solve. And the rest, my friend asked me to solve but  only can find the root $(x,y,z)=(1,1,1)$. What about the solution? Help me, thanks

The hint for the second system.
Let $x=\tan\alpha,$ $y=\tan\beta$ and $z=\tan\gamma$.
Thus, since $$\tan3t=\frac{3\tan{t}-\tan^3t}{1-3\tan^2t},$$ we need to solve the following system.
$$\tan\alpha=\tan3\gamma,$$
$$\tan\beta=\tan3\alpha$$ and
$$\tan\gamma=\tan3\beta.$$
Can you end it now?
The hint for the first system.
Rewrite the second equation in the following form.
$$x(x^2+y^2+1)=z(x^2+y^2+1).$$
