We have two coupled linear matrix equations in $\mathrm X_1, \mathrm X_2 \in \mathbb R^{n \times n}$
$$\begin{array}{rl} \alpha \, \mathrm A \mathrm X_1 + \mathrm X_1 \mathrm B &= \gamma \, \mathrm C \mathrm X_2\\ \beta \, \mathrm A \mathrm X_2 + \mathrm X_2 \mathrm B &= \delta \, \mathrm C \mathrm X_1\end{array}$$
Vectorizing both sides of both matrix equations, we obtain the following homogeneous linear system
$$\begin{bmatrix} \left((\mathrm I_n \otimes \alpha \, \mathrm A) + (\mathrm B^\top \otimes \mathrm I_n)\right) & - \mathrm I_n \otimes \gamma \, \mathrm C\\ - \mathrm I_n \otimes \delta \, \mathrm C & \left((\mathrm I_n \otimes \beta \, \mathrm A) + (\mathrm B^\top \otimes \mathrm I_n)\right) \end{bmatrix} \begin{bmatrix} \mbox{vec} (\mathrm X_1)\\ \mbox{vec} (\mathrm X_2)\end{bmatrix} = \begin{bmatrix} 0_{n^2}\\ 0_{n^2}\end{bmatrix}$$
One solution is $\mathrm X_1 = \mathrm X_2 = \mathrm O_n$. The rank of the $2n^2 \times 2n^2$ block matrix above tells us whether this solution is unique or not.