In this paper, we present a comparison theorem for the solutions of two generalized algebraic Riccati equations (GAREs) coming from two different systems. We show that the so-called strong solutions, whose related matrix pencils have all their finite eigenvalues in the closed left half plane, are maximal. The results obtained generalize the existing monotonicity results of algebraic Riccati equations. As an application of the above results, we provide a parameterization of all strong solutions of the GARE related to the singular spectral factorization of a proper transfer function with finite and infinite imaginary axis zeros.