Given matrices $A, B, C, E, F \in \mathbb{R}^{n \times n}$, where $E \succ 0$, classify matrices $A, B, C, E, F$ such that matrices $X, Y \in \mathbb{R}^{n \times n}$ are unique solutions for the following equation.
\begin{aligned} &XFY+XA+BY+C=0\\ &XY+E=0 \end{aligned}
Since $E$ is invertible, $X,Y$ too. Thus we may write the system in the unknowns $X,Y$ as follows
$Y=-X^{-1}E$ and $-XF+XAE^{-1}X-B+CE^{-1}X=0$.
Notice that the second equation is the GENERAL standard Riccati equation
$(*)$ $XPX+XQ+RX+S=0$ (choose $E$ symm. $>0$, $F=-Q,A=PE,C=RE,B=-S$).
Generically (randomly choose $P,Q,R,S$), $(*)$ admits $\binom{2n}{n}$ complex solutions. Here, you consider only the real solutions and you want only one. It's very difficult to find a NSC for the equation to admit a single real solution. It's much easier to give a sufficient condition.
Consider, for example, the CAR Equation
$A^TX+XA-XBR^{-1}B^TX+Q=0$
Though generally this equation can have several solutions, it is usually specified that we want to obtain the unique stabilizing solution, if such a solution exists.
Example with a unique complex solution (which is real!)
$X\begin{pmatrix}0&1\\0&1\end{pmatrix}X+\begin{pmatrix}-1&2\\0&-1\end{pmatrix}X-\begin{pmatrix}-3&5\\-2&2\end{pmatrix}=0$.