Given $k\geq 9$, let us write $k=2p$ or $k=2p+1$ depending on the parity of $k$, and let $\mu \in (1/2,\infty)$. Then the $k^{\text{th}}$ generalized Gromov width of $M_{\mu}^{1}$ is given by: $${\mathsf{w}}_{2p}(M_{\mu}^{1}) =
  \begin{cases}
    c_{\operatorname{vol}}=\sqrt{\frac{2\mu+1}{2p}} & \text{~if~} \mu\in \left[1/2,~ p-\sqrt{2p}\right)\\
    \frac{p+\mu}{2p} & \text{~if~} \mu\in\left[p-\sqrt{2p}~ , p\right) \\
    1 & \text{~if~} \mu\in\left[p,~\infty\right)\\
  \end{cases}$$ and $${\mathsf{w}}_{2p+1}(M_{\mu}^{1}) =
  \begin{cases}
    c_{\operatorname{vol}}=\sqrt{\frac{2\mu+1}{2p+1}} & \text{~if~} \mu\in \left[1/2,~ \frac{p^{3}-2p^{2}+1-(p-1)\sqrt{2p+1}}{p^{2}}\right)\\
    \frac{p(p+\mu-1)}{2p^{2}-p-1} & \text{~if~} \mu\in \left[\frac{p^{3}-2p^{2}+1-(p-1)\sqrt{2p+1}}{p^{2}},~ \frac{p(p-1)}{p+1}\right)\\
    \frac{p+\mu}{2p} & \text{~if~} \mu\in\left[\frac{p(p-1)}{p+1} ,~ p\right) \\
    1 & \text{~if~} \mu\in\left[p,~\infty\right)\\
  \end{cases}$$

The previous results allow us to compute the stability numbers of the nontrivial bundle, namely

\[cor:MainPackingNumbersTwisted\] The stability number $N_{{\rm stab}}$ of $M_{\mu}^{1}$ is $$N_{{\rm stab}}=
\begin{cases}
8 & \text{~if~} \mu\in  \mathcal{S} \\
9 & \text{~if~} \mu\in   \left(0,~\frac{3}{2}\right) \\
N_{{\rm even}}-1  & \text{~if~} \mu\in \left[\frac{3}{2},~\infty \right)
\end{cases}$$

To conclude this introduction, we want to point out that our presentation is written with a dual purpose in mind. First, we obtained our results through an hybrid process of mathematical reasoning and computer-aided symbolic computations using SAGE [@sage], and our exposition replicates part of that process. Secondly, although we don’t discuss the arithmetic aspects of the reduction process in the present paper, we advance the idea that the reduction algorithm is an 