I'm trying to find the root $\mathbf{f(x)=0}$ to the following sets of equations
$$ f_1(x,y,z) = x^\prime - \frac{x}{\sqrt{x^2+y^2+z^2}} = 0 \\ f_2(x,y,z) = y^\prime - \frac{y}{\sqrt{x^2+y^2+z^2}} = 0 \\ f_3(x,y,z) = z^\prime - \frac{z}{\sqrt{x^2+y^2+z^2}} = 0 $$
where $x^\prime,y^\prime,z^\prime$ are known. When I try to use the Newton method, i.e $\mathbf{x}^k = \mathbf{x}^{k-1} - J^{-1} \mathbf{f(x^{k-1})}$ it fails since the jacobian $J$ is not invertible (no unique solution). The root $(x,y,z)$ I want to find lies in a plane $ax+by+cz=d$. How do I limit the search to this subspace of $\mathbf{R}^3$?