I am trying to find non-trivial solutions for $AX+XA=0$, where $A$ and $X$ are real, symmetric square matrices. $A$ is given and positive definite. I want to solve for $X$, which should be positive definite or positive semi-definite. Obviously, $X=0$ solves this equation, but I want to find a solution $X\neq0$.
The equation $AX+XA+Q=0$ is known in as the continuous Lyapunov equation in control theory and there is a theorem on existence and uniqueness of solutions $X$ if $Q$ is positive definite. However, in my case $Q=0$, so the theorem does not apply to my problem.
Are there any results on the existence of non-trivial $X$ and/or how to find them?
Of course the theorem does not apply when $Q=0$ because there is only the solution $X=0_n$ in $M_n$.
Proof. Let $(\lambda_i>0)_i$ be the spectrum of $A$. The eigenvalues of the linear function $f:X\in M_n\rightarrow AX+XA\in M_n$ are the $(\lambda_i+\lambda_j>0)_{i,j}$. Then $f$ is a bijection and the sole solution in $M_n$ is $0_n$.