Analytic perturbation of Sylvester matrix equations

We consider an analytic perturbation of the Sylvester matrix equation. We are mainly interested in the singular case, that is, when the null space of the unperturbed Sylvester operator is not trivial, but the perturbed equation has a unique solution. In this case, the solution of the perturbed equation can be given in terms of a Laurent series. We provide a necessary and sufficient condition for the existence of a Laurent series with a first-order pole. A recursive procedure for the calculation of the Laurent series' coefficients is given.