  We study the interplay between electron-electron interaction and a
Jahn-Teller phonon coupling in a two-orbital Hubbard model. We demonstrate that
the e-ph interaction coexists with the Mott localization driven by the Hubbard
repulsion U, but it competes with the Hund's coupling J. This interplay leads
to two spectacularly different Mott insulators, a standard high-spin Mott
insulator with frozen phonons which is stable when the Hund's coupling
prevails, and a low-spin Mott-bipolaronic insulator favoured by phonons, where
the characteristic features of Mott insulators and bipolarons coexist. The two
phases are separated by a sharp boundary along which an intriguing intermediate
solution emerges as a kind of compromise between the two solutions.
