Given the matrix $Ainmathbb{R}^{ntimes n}$ with all eigenvalues inside the unit circle and the symmetric positive definite matrix $Pinmathbb{R}^{ntimes n}$ satisfying $ APA^top-P+I=0 $, I need to find the set of solutions $S$ for

$$

S P^{-1} S^top + AS^top + S A^top leq 0

$$

It is obvious that this set of solutions is nonempty since $S=0$ satisfies the above relation. Also another trivial solution is $S=-A P$ as well as $S= -alpha v v^top$ for some values of $alpha>0$, where $v$ is an eigenvector of $A$. However, I wanted to see if there is any more general form of the solution $S$.