# systems of equations – Set of solutions for given inequality

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 $$S$$.