  Theoretical description of reactions in general, and the theory for $(d,p)$
reactions, in particular, needs to advance into the new century. Here deuteron
stripping processes off a target nucleus consisting of ${A}$ nucleons are
treated within the framework of the few-body integral equations theory. The
generalized Faddeev equations in the AGS form, which take into account the
target excitations, with realistic optical potentials provide the most advanced
and complete description of the deuteron stripping. The main problem in
practical application of such equations is the screening of the Coulomb
potential, which works only for light nuclei. In this paper we present a new
formulation of the Faddeev equations in the AGS form taking into account the
target excitations with explicit inclusion of the Coulomb interaction. By
projecting the $(A+2)$-body operators onto target states, matrix three-body
integral equations are derived which allow for the incorporation of the excited
states of the target nucleons. Using the explicit equations for the partial
Coulomb scattering wave functions in the momentum space we present the AGS
equations in the Coulomb distorted wave representation without screening
procedure. We also use the explicit expression for the off-shell two-body
Coulomb scattering $T$-matrix which is needed to calculate the effective
potentials in the AGS equations. The integrals containing the off-shell Coulomb
T-matrix are regularized to make the obtained equations suitable for
calculations. For $NN$ and nucleon-target nuclear interactions we assume the
separable potentials what significantly simplifies solution of the AGS
equations.
