  We investigate the interplay of phonons and correlations in superconducting
pairing by introducing a model Hamiltonian with on-site repulsion and couplings
to several vibration branches having the Cu-O plane of the cuprates as a
paradigm. We express the electron-phonon coupling (EP) through two force
constants for O-Cu and O-O bond stretchings. Without phonons, this reduces to
the Hubbard Model, and allows purely electronic W=0 pairing. A W=0 pair is a
two-body singlet eigenstate of the Hubbard Hamiltonian, with no double
occupancy, which gets bound from interactions with background particles.
Indeed, this mechanism produces a Kohn-Luttinger-like pairing from the Hubbard
repulsion, provided that its symmetry is not severely distorted.
  From the many-body theory, a canonical transformation extracts the effective
two-body problem, which lends itself to numerical analysis in case studies. As
a test, we use as a prototype system the $\cu$ cluster. We show analytically
that at weak EP coupling the additive contributions of the half-breathing modes
reinforce the electronic pairing. At intermediate and strong EP coupling and $U
\sim t$, the model behaves in a complex and intriguing way.
