  In this paper we prove the existence of quasi-periodic, small-amplitude,
solutions for quasi-linear Hamiltonian perturbations of the non-linear
Schroedinger equation on the torus in presence of a quasi-periodic forcing. In
particular we prove that such solutions are linearly stable. The proof is based
on a Nash-Moser implicit function theorem and on a reducibility result on the
linearized operator in a neighborhood of zero. The proof of the reducibility
relies on changes of coordinates such as diffeomorphisms of the torus,
pseudo-differential operators and a KAM-reducibility arguments. Due to the
multiplicity of the eigenvalues we obtain a block-diagonalization.
