Lyapunov matrix equation theorem

Question

I know the following Lyapunov's theorem:

For any symmetric positive definite matrix $Q$, the following are equivalent

1- The matrix equation $A^T X + X A = -Q $ has a unique solution $X$ that is symmetric and positive definite.

2- Matrix $A$ is Hurwitz, i.e. all of its eigenvalues have strictly negative real parts.

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My question is, can I propose this following theorem:

If matrix $A$ is not Hurwitz, then $A^T X + X A = -I $ does not have a solution.

Thank you.

Answer 1

No. A counterexample: $$ A=\left(\begin{array}{rr} -2&0\\ 0&1\\ \end{array}\right) \qquad X=\left(\begin{array}{rr} \frac14&0\\ 0&-\frac12\\ \end{array}\right). $$