by ${\breve{\mathfrak{E}}_{\vec{k},\infty}}$. It turns out that $$\left({\breve{\mathfrak{E}}_{\vec{k},\infty}}\right)_{(\alpha,\beta)}\simeq {\mathcal{E}}_{\alpha,\beta,\infty}
\label{eqdEi}$$ for all points $(\alpha,\beta)\in{L_{\vec{k}}}$.

Let $\mathfrak N_{\vec k}$ denote the direct image $\breve{\mathfrak u}_{2\ast} \breve{\mathfrak E}_{\vec k,\infty}$. Since $\breve{\mathfrak E}_{\vec k,\infty}$ is a trivial vector bundle on each fibre of $\breve{\mathfrak u}_{2}$, the sheaf $\mathfrak N_{\vec k}$ is locally free of rank $r$. Let $P_{\vec k}$ be its bundle of linear frames, \[pagPk\] which is a principal $\operatorname{GL}(r)$ bundle on $L_{\vec k}$, whose fibre at $(\alpha,\beta)\in{L_{\vec{k}}}$ may be identified with the bundle of linear frames of the vector bundle $\mathcal E_{\alpha,\beta,\infty}$. Moreover, if $$\label{tau} 
\tau\colon P_{\vec k} \to L_{\vec k}$$ is the projection, the vector bundle $\widetilde{\mathfrak N}_{\vec k}=\tau^\ast \mathfrak N_{\vec k}$ is trivial. Note that a framed sheaf $({\mathcal{E}},\theta)$ determines a point of ${P_{\vec{k}}}$. Indeed the monad whose cohomology is ${\mathcal{E}}$ gives a point $(\alpha,\beta)\in{L_{\vec{k}}}$, and the framing $\theta$ gives a point of ${P_{\vec{k}}}$ in the fibre over $(\alpha,\beta)$.

The family $\left({\widetilde{\mathfrak{E}}_{\vec{k}}},{\widetilde{\Theta}_{\vec{k}}}\right)$ {#sectEtT}
---------------------------------------------------------------------------------------------

The geometrical environment of this subsection is provided by the varieties ${\Sigma_n\times{P_{\vec{k}}}}$ and ${{\ell_\infty}\times{P_{\vec{k}}}}$, together with the canonical projections shown in the following diagram: $$\xymatrix{
{\ell_\infty}\ar@{^{(}->}[r] & \Sigma_n\\
{{\ell_\infty}\times{P_{\vec{k}}}}\ar@{^{(}->}[r] \ar[u]_-{{\tilde{\mathfrak{u}}}_1} \ar[dr]_{{\tilde{\mathfrak{u}}}_2} & {\Sigma_n\times{P_{\vec{k}}}}\ar[u]_{{\tilde{\mathfrak{t}}}_1} \ar[d]^{{\tilde{\mathfrak{t}}}_2}\\
 & {P_{\vec{k}}}\,.
}$$

Let ${\widetilde{\mathfrak{E}}_{\vec{k}}}:=\left({\operatorname{id}}_{\Sigma_n}\times\tau\right)^*{\breve{\mathfrak{E}}_{\vec{k}}}$. This sheaf satisfies condition $\vec{k}$, and in 