arXiv:2610.30804v1 Announce Type: cross Abstract: This paper develops projection methods for computing low-rank solutions to large-scale coupled generalized Sylvester equations. The dominant subspaces of these solutions are shown to be spanned by specific resolvent directions, motivating a Petrov--Galerkin projection framework. Conditions for the unique solvability of the projected coupled generalized Sylvester equations are also discussed. Based on these dominant subspaces, two algorithms are proposed: an extended Krylov-based projection algorithm and a rational Krylov-based projection algorithm. The former uses fixed shifts, whereas the latter uses adaptive shifts. Numerical experiments on large-scale benchmark problems demonstrate that the proposed methods achieve high accuracy and fast convergence.