Well, the Riccati matrix equation and its variants transform (assuming the leading "coefficient" is invertible) to the quadratic $Z^2 + AZ + B = 0$ (where $Z$ is the unknown matrix and $A,B$ are square). There are generically $C(2n,n)$ solutions, although there could be less, or a continuum (but not greater than $C(2n,n)$ and less than $c$); if $A$ and $B$ commute, the generic number is $2^n$.
One reference is,
D Handelman [me], Fixed points of two-sided fractional matrix transformations, Fixed point theory and its applications, 2007, ID41930, doi:10.1155/2007/41930,
which is freely downloadable.
It's mainly concerned with fixed points of the densely-defined transformation (on $n \times n$ matrices) $X \mapsto (I - CXD)^{-1}$ (where $C$ and $D$ are given), which can be transformed into the quadratic above, and variations on it. It turns out there is a natural graph structure on the solutions, which is generically the Johnson graph.
Edit: Oops, on rereading your question, I see that you required the coefficients to be scalars, which is much more tractible. Oh well, never mind. (If you remember Gilda Radner on SNL ....)