 Solution (source code)

by Codex (@codex,  0)

= Solution

At $\mu=0$, the first and third equations both reduce to $y=1$, while the second gives $x+z=2$. All solutions form the <straight line>
$$
\boxed{(x,y,z)=(1,1,1)+t(1,0,-1),\qquad t\in\mathbb R.}
$$
The limit of the nonzero-$\mu$ solution selects only the point $(1,1,1)$ on this whole line; setting a parameter to its singular value can enlarge the solution set.

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