<s>[INST] Can't solve non linear system I'm trying to solve this system:
$\begin{cases}(1+y)(1+z) = 2\lambda x \\ (1+x)(1+z) = 2\lambda y \\ (1+x)(1+y)= 2\lambda z \\ x^2+y^2+z^2 - 1 = 0\end{cases}$
We can assume that $\lambda \neq 0$
How would you solve this? No matter in which direction I go it seems unpleasant.
 [/INST] We have 
$$
\left\{
\begin{array}{rcl}
(1+x)(1+y)(1+z) &=&2\lambda x(1+x)\\
(1+x)(1+y)(1+z) &=&2\lambda y(1+y)\\
(1+x)(1+y)(1+z) &=&2\lambda z(1+z)\\
x^2+y^2+z^2&=&1
\end{array}
\right.
$$
Note in this case that if we switch as unknowns $x$, $y$ and $z$ with each other the system does not change. It is suggested that one of the solutions is $x=y=z$. Then $x^2+y^2+z^2=1$ and $x=y=z$ implies $3x^2=1$, $3y^2=1$, $3z^2=1$, that is, $x=\pm\frac{1}{\sqrt{3}}$, $y=\pm\frac{1}{\sqrt{3}}$ and $z=\pm\frac{1}{\sqrt{3}}$.
</s>