This paper considers the generalized algebraic Riccati equation (GARE) associated with a non-standard J-spectral factorization problem (NS-J-SFP) for descriptor systems, of which system matrices have full column rank. We derive necessary and sufficient conditions for the existence of a semi-stabilizing solution of the GARE based on the Hamiltonian matrix pencil approach and gives a parametrization of all the semi-stabilizing solutions. Then we provide a nonstandard J-spectral factor (NS-J-SF) in terms of a particular semi-stabilizing solution of the GARE. A simple numerical result is also included.