On hyperbolicity of SU(2)-equivariant, punctured disc bundles over the complex affine quadric

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.