It is called a **T-Sylvester equation**, or *-Sylvester equation in the complex case. Solvability conditions and a pseudocode algorithm based on the Schur form are in 	https://doi.org/10.13001/1081-3810.1479  . Analogous considerations for a more general class of equations and a Fortran-90 implementation of the last step of the resulting solution algorithm (the back-substitution on the triangular version of the equation) are in my paper  https://doi.org/10.1002/nla.2261 . I don't think you will find something in Slicot, because it has no immediate control theory applications.