  We compare and contrast the entanglement in the ground state of two
Jahn-Teller models. The $E\otimes\beta$ system models the coupling of a
two-level electronic system, or qubit, to a single oscillator mode, while the
$E\otimes\epsilon$ models the qubit coupled to two independent, degenerate
oscillator modes. In the absence of a transverse magnetic field applied to the
qubit, both systems exhibit a degenerate ground state. Whereas there always
exists a completely separable ground state in the $E\otimes\beta$ system, the
ground states of the $E\otimes\epsilon$ model always exhibit entanglement. For
the $E\otimes\beta$ case we aim to clarify results from previous work, alluding
to a link between the ground state entanglement characteristics and a
bifurcation of a fixed point in the classical analogue. In the
$E\otimes\epsilon$ case we make use of an ansatz for the ground state. We
compare this ansatz to exact numerical calculations and use it to investigate
how the entanglement is shared between the three system degrees of freedom.
