I have the following vector equation: $$ {\bf Ax} + {\bf b} + {\bf Cx}^{ \circ - 1} = {\bf 0}_n $$ Where ${\bf x}$ is a vector of unknown variables. ${\bf b},{\bf x}, {\bf x}^{\circ - 1}, {\bf 0}_n \in \mathbb{R}^{n}$; ${\bf A},{\bf C} \in \mathbb{R}^{n \times n}$ and $\bf C$ is a diagonal matrix. ${\bf x}^{\circ - 1}$ is the Hadamard (pointwise) product inverse of ${\bf x}$, that means ${\bf x} \circ {\bf x}^{\circ - 1} = {\bf 1} \in \mathbb{R}^{n}$. I thought about converting the equation into a quadratic vector equation and ended with something like this: $$ {\bf x} \circ \left({\bf Ax}\right) + {\bf Bx} + {\bf c} = {\bf 0}_n $$ Where ${\bf B} \in \mathbb{R}^{n \times n}$ is a diagonal matrix whose diagonal entries are the same of ${\bf b}$ and ${\bf c} \in \mathbb{R}^{n}$ is a vector whose entries are equal to the diagonal entries of ${\bf C}$. I have not progressed any further than that. Is there any way of analytically solving this equation? whether by doing some kind of "completing the square" or any other method.
NOTE: If necessary it can be assumed that the entries of ${\bf A}$ and ${\bf C}$ are non-negative, and that ${\bf A}$ and ${\bf C}$ are invertible.
EDIT: Another equivalent equation is: $$ \left( {\bf X}^T{\bf AX} + {\bf X}^T{\bf B} + {\bf C} \right) {\bf 1}_n = {\bf 0}_n $$ Where ${\bf 1}_n \in \mathbb{R}^{n}$ is a vector of $1$'s and ${\bf X}$ is a diagonal matrix whose diagonal entries are the same of ${\bf x}$.
EDIT2: Second equivalent equation is: $$ \left({\bf Ax}\right) \circ \left({\bf Ax}\right) + {\bf ABx} + {\bf Ac} = {\bf 0}_n $$ But still, no progress have been made.