  The Sylvester equation $AX-XB=C$ is considered in the setting of quaternion
matrices. Conditions that are necessary and sufficient for the existence of a
unique solution are well-known. We study the complementary case where the
equation either has infinitely many solutions or does not have solutions at
all. Special attention is given to the case where $A$ and $B$ are respectively,
lower and upper triangular two-diagonal matrices (in particular, if $A$ and $B$
are Jordan blocks)
