I have seen outlined in [this comment os mathoverflow](https://mathoverflow.net/a/145251/97814) how to solve quadratic matrix equations of the form
$$
XCX + AX = I
$$
where $X \in \mathbb{R}^{n\times n}$, $C = C^T > 0 \in \mathbb{R}^{n\times n}$, and $I$ is identity of corresponding size.
I need to look this up more thoroughly; thus I need a name of a related theorem or a book, paper or lecture to do some advanced reading about conditions for existence and uniqueness of solutions and better even, under what conditions does a positive definite $X$ exist.



Source of equation - theorems about solving quadratic matrix equations
Bini, Iannazzo, Meini, [*Numerical Solution of algebraic Riccati equations*](http://bookstore.siam.org/FA09/), SIAM books, seems a good starting point to me. It is a monograph that deals both with the symmetric and the non-symmetric case and assumes no previous knowledge in control theory.


