  Taking into account microscopic properties of most usual high-$T_{c}$
superconductors, like cuprates, we define a class of microscopic model
Hamiltonians for two fermions (electrons or holes) and one boson (bipolaron) on
the two-dimensional square lattice. We establish that these model Hamiltonians
can show d-wave paring at the bottom of their spectrum, despite their space
isotropy. This phenomenon appear when a "giant electron-phonon anomaly" is
present at the boundaries of the Brillouin zone ("half breathing"
bond-stretching mode), like in doped cuprates. Our results can be used to
derive effective electron-electron interactions mediated by bipolarons and we
discuss regimes where the corresponding model is relevant for the physics of
high-temperature superconductivity and can be mathematically rigorously
studied.
