On a relationship between the T-congruence Sylvester equation and the Lyapunov equation

We consider the T-congruence Sylvester equation $AX+X^{\rm T}B=C$, where $A\in \mathbb R^{m\times n}$, $B\in \mathbb R^{n\times m}$ and $C\in \mathbb R^{m\times m}$ are given, and matrix $X \in \mathbb R^{n\times m}$ is to be determined. The T-congruence Sylvester equation has recently attracted attention because of a relationship with palindromic eigenvalue problems. For example, necessary and sufficient conditions for the existence and uniqueness of solutions, and numerical solvers have been intensively studied. In this note, we will show that, under a certain condition, the T-congruence Sylvester equation can be transformed into the Lyapunov equation. This may lead to further properties and efficient numerical solvers by utilizing a great deal of studies on the Lyapunov equation.