in equation .

Recall by Theorem \[master\](a) that a $\lambda$-bracket on ${\mathcal{V}}_N^\infty$ is uniquely determined by assigning the $\lambda$-brackets on generators, $\{{u_i}_\lambda{u_j}\}$, for all $i,j\in I$, or, equivalently, their generating series $\{L(z)_\lambda L(w)\}$. In particular, there exists a unique $\lambda$-bracket on ${\mathcal{V}}_N^\infty$ such that $L(\partial)-c$ is of Adler type: $$\label{generating2}
\{L(z)_\lambda L(w)\}_c=H^{(L-c)}(\lambda)(w,z)\,.$$

It is clear from the definition of the Adler map $A^{(L)}$ that $A^{(L-c)}$ is linear in $c$, and therefore $$\label{HK}
H^{(L-c)}(\partial)=H(\partial)-cK(\partial)\,,$$ where $H=H^{(L)}$ is given by and , and $K\in\operatorname{Mat}_{I\times I}{\mathcal{V}}_N^\infty[\partial]$.

\[L-c\] The 1-parameter family of $\lambda$-brackets $\{\cdot\,_\lambda\,\cdot\}_c=\{\cdot\,_\lambda\,\cdot\}_H-c\{\cdot\,_\lambda\,\cdot\}_K$, $c\in{\mathbb{F}}$, defines a structure of bi-PVA on ${\mathcal{V}}_N^\infty$.

The statement is a special case of Remark \[subalgebra\]. By Definition \[hamop\], we have to prove that the $\lambda$-bracket on ${\mathcal{V}}$ defined by equation defines a PVA structure on ${\mathcal{V}}$. By Theorem \[master\], this is the same as proving skew-symmetry and Jacobi identity on generators, which are the same as equations and respectively.

\[agd\] The *AGD bi-PVA* ${\mathcal{V}}_N^\infty$ is given by the 1-parameter family of $\lambda$-brackets $\{\cdot\,_\lambda\,\cdot\}_c$, $c\in{\mathbb{F}}$. The operators $K$ and $H$ are usually called, respectively, the *first* and the *second Adler-Gelfand-Dickey Poisson structures*.

Using equation we get the following explicit formulas for $\{L(z)_\lambda L(w)\}_c
=\{L(z)_\lambda L(w)\}_H-c\{L(z)_\lambda L(w)\}_K$: $$\label{eq:H}
\begin{array}{c}
\vphantom{\Big(}
