  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.
