Given the matrix $A\in\mathbb{R}^{n\times n}$ with all eigenvalues inside the unit circle and the symmetric positive definite matrix $P\in\mathbb{R}^{n\times n}$ 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 $S$.