I happen to be working on a problem that reduces to solving the following equation:

$$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$

where A through D are known matrices ( A, B, D are real, symmetric matrices and C is real and antisymmetric), and X is an unknown square matrix to be solved for.

Is there a name for this equation, and is there any known algorithm for solving this equation? (Without the C X C term this reduces to the continuous Lyapunov equation given either A or B is an invertible matrix. I wonder if anyone working in control theory may have seen such equations before.)