  We discuss the ground state entanglement of the $E\otimes\epsilon$
Jahn-Teller model in the presence of a strong transverse magnetic field as a
function of the vibronic coupling strength. A complete characterization is
given of the phenomenon of entanglement sharing in a system composed by a qubit
coupled to two bosonic modes. Using the residual $I$-tangle, we find that
three-partite entanglement is significantly present in the system in the
parameter region near the bifurcation point of the corresponding classical
model.
