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.)