  In this paper, we extend the GKZ-system method to the more general case:
compact Complete Intersection Calabi-Yau manifolds (CICY). For several
one-deformation modulus compact CICYs with D-branes, the on-shell
superpotentials in this paper from the extended GKZ-system method are exactly
consistent with published results obtained from other methods. We further
compute the off-shell superpotentials of these models. Then we obtain both the
on-shell and off-shell superpotentials for sev- eral two-deformation moduli
compact CICYs with D-branes by using the extended GKZ-system method. The
discrete symmetrical groups, Z2, Z3 and Z4, of the holo- morphic curves wrapped
by D-branes play the important roles in computing the superpotentials, in some
sense, they are the quantum symmetries of these models. Furthermore, through
the mirror symmetry, the Ooguri-Vafa invariants are extracted from the A-model
instanton expansion.
