Let $n \in \mathbb{N}$. Let $A,B,C$ real matrices of size $n \times n$. Let $\alpha,\beta,\gamma,\delta \in \mathbb{R}^{4}$ such that $\alpha,\beta$ are non-zero. 


I an looking for two matrices $T\_1$ and $T\_2$ belonging to $\mathbb{R}^{n \times n}$ such that:
\begin{eqnarray}
\alpha A T\_1 + T\_1 B &=& \gamma C T\_2 \\ 
\beta A T\_2 + T\_2 B &=& \delta C T\_1
\end{eqnarray}


1. Is there any known result on when these coupled Sylvester equations admit a solution?
2. If there is a solution, when is it unique (as a function of the scalars $\alpha,\beta,\gamma,\delta$) ?


Edit: as was mentioned by Carlo Beenakker, the equations admit the trivial solution $T\_1 = T\_2 = 0$. What if now, we assume both $T\_1$ and $T\_2$ non-singular?



Coupled Sylvester equations
There is a simpler criterion that does not involve constructing a $2n^2 \times 2n^2$ matrix and checking its singularity (which is a costly operation).


Your equation is equivalent to the standard Sylvester equation (with rectangular unknown)
$$
M \begin{bmatrix}T\_1\\T\_2\end{bmatrix} + \begin{bmatrix}T\_1\\T\_2\end{bmatrix} B = 0,
$$
where
$$M=
\begin{bmatrix}
\alpha A & -\gamma C\\
-\delta C & \beta A
\end{bmatrix},
$$
so you can rely on all the existing theory for the [Sylvester equation](https://en.wikipedia.org/wiki/Sylvester_equation). For instance, a classical result is: there is a nonzero solution to the equation iff $M$ and $-B$ have no common eigenvalues.


