The continuous Lyapunov equation is normally formulated as: $AX + XA^T + Q = 0$. Given the matrices $A$ and $Q$, there exists a unique $X$ iff the linear system described by $A$ is globally asymptotically stable. [EDIT: note that $Q$ must also be positive ($Q>0$) and symmetric]
However, I'm wondering if it's possible to solve instead for the system itself. In other words, given the matrices $X$ and $Q$, does a unique $A$ exist? Is there a way to find it?