  The phase diagram of half-doped manganite systems of formula
A_{0.5}A'_{0.5}MnO_3 is investigated within a single-orbital model
incorporating magnetic double-exchange and superexchange, together with
intersite Coulomb and electron-lattice interactions. Strong Jahn-Teller and
breathing mode deformations compete together and result in shear lattice
deformations. The latters stabilize the charge-ordered CE-type phase, which
undergo first-order transitions with temperature or magnetic field to either
Ferromagnetic metallic or Paramagnetic insulating phases. An essential feature
is the self-consistent screening of Coulomb and electron-phonon interactions in
the ferromagnetic phase.
