An analysis of the matrix equation $AX+\overline{X}B=C$

The matrix equation $AX+\overline{X}B=C$, where $\overline{X}B$ is obtained by the entry-wise conjugation of $X$, is examined. On the basis of analogy with the Sylvester matrix equation, special cases are distinguished where the former equation corresponds to normal and self-adjoint Sylvester equations. Efficient numerical algorithms are proposed for these special cases.