For what conditions on $A$, $B$ and $C$ (square matrices of size $n$) would there be a unique solution to
$$
ABX + AXC + XBC = D,
$$
for any $D$? Can one expect a characterization similar to the Sylvester Theorem, which states that there always exists a unique solution to $AX + XB = C$, for any $C$, if and only if $A$ and $-B$ do not share an eigenvalue? And then, can this be extended to equations of the form $ABCX + ABXD + AXCD + XBCD = E$, and beyond?