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}

- Is there any known result on when these coupled Sylvester equations admit a solution?
- 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?