The phase diagram of half-doped manganite systems of formula A(0.5)A ' 0.5M
nO3 is investigated within a single-orbital model incorporating magnetic do
uble-exchange and superexchange, together with intersite, Coulomb and elect
ron-lattice interactions. Strong Jahn-Teller and breathing mode deformation
s compete together and result in shear lattice deformations. The latter sta
bilize the charge-ordered CE-type phase, which undergo first-order transiti
ons 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.