  Given a holomorphic line bundle over the complex affine quadric $Q^2$, we
investigate its Stein, SU(2)-equivariant disc bundles. Up to equivariant
biholomorphism, these are all contained in a maximal one, say $\Omega_{max}$.
By removing the zero section to $\Omega_{max}$ one obtains the unique Stein,
SU(2)-equivariant, punctured disc bundle over $Q^2$ which contains entire
curves. All other such punctured disc bundles are shown to be Kobayashi
hyperbolic.
