I have seen outlined in this comment os mathoverflow 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.
Multiplying by $C$ from the right, the equation is reduced to $$Y^2 + AY - C = O,$$ where $Y=XC$. Solution to such polynomial matrix equations is described in Chapter VIII in F.R.Gantmachers. The Theory Of Matrices.