  This work is to provide a comprehensive treatment of the relationship between
the theory of the generalized (palindromic) eigenvalue problem and the theory
of the Sylvester-type equations. Under a regularity assumption for a specific
matrix pencil, we show that the solution of the $\star$-Sylvester matrix
equation is uniquely determined and can be obtained by considering its
corresponding deflating subspace. We also propose an iterative method with
quadratic convergence to compute the stabilizing solution of the
$\star$-Sylvester matrix equation via the well-developed palindromic doubling
algorithm. We believe that our discussion is the first which implements the
tactic of the deflating subspace for solving Sylvester equations and could give
rise to the possibility of developing an advanced and effective solver for
different types of matrix equations.
