<s>[INST] New Question: Find $x,y,z$ s.t. $x+y+z=1$ and $x^2+y^2+z^2=1$, and $x,y,z \in ( - 1,0) \cup (0,1)$ If in my earlier question [1], we relieve the constraints such that $x,y,z$ can also get values between $(−1,0)$, it seems that the system has a solution. Now, the question is how we can find this solution? 
$$\begin{cases}&x + y + z = 1\\
&{x^2} + {y^2} + {z^2} = 1\\
&x \ne y \ne z, x \ne z\\
&x,y,z \in ( - 1,0) \cup (0,1)
 \end{cases}$$
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[1] Multivariate-Multi-objective Optimization Problem: $x+y+z = 1$ and $x^2+y^2+z^2 = 1$
 [/INST] The set of solutions become a circle which intersect of next two spheres. 
$$(x-1/3)^2+(y-1/3)^2+(z-1/3)^2=2/3$$
$$x^2+y^2+z^2=1$$ except $x=y,y=z,z=x$.
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