We represent the Poincaré dual of $[{\omega}_{\mu,c}^{0}]$ by the vector $$\label{eq:InitialVectorTrivial}
  v_{\mu,c}^{0} :=
  \left(\mu+1-c\,; \mu-c, c^{\times (k-1)}, 1-c\right)\in H_{2}(X_{k+1};{\mathbb{R}})$$

Similarly, we identify the $k$-fold blow-ups of $M^1_{\mu}$ of equal sizes $c=c_{1}=\cdots = c_{k}$, with ${\mathbb{CP}}^{2}$ blown up $(k+1)$ times endowed with a symplectic form ${\omega}_{\mu,c}^{1}$ which gives area $\mu+1$ to a line and areas $\{\mu, c,\ldots, c\}$ to the $(k+1)$ exceptional divisors. The Poincaré dual of $[{\omega}_{\mu,c}^{1}]$ is thus represented by the vector $$\label{eq:InitialVectorTwisted}
  v_{\mu,c}^{1} :=
  \left(\mu+1\,; \mu, c^{\times k}\right)\in H_{2}(X_{k+1};{\mathbb{R}})$$

The existence of an embedding of $k$ disjoint balls of capacity $c$ into $M^i_{\mu}$ is then equivalent to $v_{\mu,c}^{i}$ belonging to the symplectic cone $\mathcal{C}_{K}$ of $X_{k+1}$, where $K:=(3\,;1^{\times (k+1)})$. Hence, given $k\geq 8$, the computation of the generalized Gromov widths ${\mathsf{w}}_{k}(M^0_{\mu})$ and of the packing numbers $p_{k}(M^0_{\mu})$ reduces to finding the largest capacity $c>0$ such that

-   $v_{\mu,c}^{i}\in\mathcal{P}_{+}$ (i.e., $v_{\mu,c}^{i}\cdot v_{\mu,c}^{i}\geq 0$);

-   the orbit of $v_{\mu,c}^{i}$ under $D_{K}(1,k+1)$ only contains non-negative vectors or, equivalently, the reduction algorithm applied to $v_{\mu,c}^{i}$ produces a reduced and non-negative vector.

A posteriori, once one knows the generalized Gromov widths given in Theorem \[thm:MainTheoremTrivial\] and Theorem \[thm:MainTheoremTwisted\], one can easily check that those numbers are the right ones. Indeed, given $\mu$ and 