  We consider generalizations of the Sylvester matrix equation, consisting of
the sum of a Sylvester operator and a linear operator $\Pi$ with a particular
structure. More precisely, the commutator of the matrix coefficients of the
operator $\Pi$ and the Sylvester operator coefficients are assumed to be
matrices with low rank. We show (under certain additional conditions) low-rank
approximability of this problem, i.e., the solution to this matrix equation can
be approximated with a low-rank matrix. Projection methods have successfully
been used to solve other matrix equations with low-rank approximability. We
propose a new projection method for this class of matrix equations. The choice
of subspace is a crucial ingredient for any projection method for matrix
equations. Our method is based on an adaption and extension of the extended
Krylov subspace method for Sylvester equations. A constructive choice of the
starting vector/block is derived from the low-rank commutators. We illustrate
the effectiveness of our method by solving large-scale matrix equations arising
from applications in control theory and the discretization of PDEs. The
advantages of our approach in comparison to other methods are also illustrated.
