TITLE: Is there a name for the matrix equation A X B + B X A + C X C = D?
QUESTION [7 upvotes]: 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.)

REPLY [2 votes]: Apart from very special cases (something commuting with something else), as far as I know there is no efficient algorithm for this kind of equations with more than two summands. (by "efficient" I mean "better than the Kronecker product approach"). 
May sound strange, but I would actually suggest solving the Kronecker product system with an iterative method like SYMMLQ, or CG if it's positive definite. Matrix-vector products cost "only" $O(n^3)$, and dropping a term provides a better-than-nothing preconditioner.