Given square matrices $A,Q$ with $Q>0$ the matrix equation known as the Lyapunov equation is given by
$$A^TX+XA+Q=0.$$
Lyapunov's theorem state that the solution $X$ is symmetric and satisfies $X>0$ if $A$ is hurwitz. Let us now consider the generalized "Sylvesters equation" for square matrices A,B,C
$$AX+XB+C=0.$$ Let us assume that $A,-B$ does not have common eigenvalues. Then there is a solution to the sylvester equation (see here)
question:
Is there an analogous criteria that can be put upon $A,B,C$ so that if $C$ is symmetric and $C>0$ then there exist a solution $X$ which is symmetric and $X>0$?