generators ($i,j,k\in I$): $$\label{jacobigen}
    \{u_i{}_\lambda\{u_j{}_\mu u_k\}_H\}_H
    -\{u_j{}_\mu\{u_i{}_\lambda u_k\}_H\}_H
    =
    \{{\{u_i{}_\lambda u_j\}_H}_{\lambda+\mu}u_k\}_H
    \,.$$

Theorem \[master\] holds in the more general situation when ${\mathcal{V}}$ is any algebra of differential functions in the variables $\{u_i\}_{i\in I}$ (see [@BDSK09] for a definition), but in the present paper we will only consider the case when ${\mathcal{V}}=R_I$.

\[hamop\] A *Poisson structure* on ${\mathcal{V}}$ is a matrix differential operator $H(\partial)\in\operatorname{Mat}_{\ell\times\ell}{\mathcal{V}}[\partial]$ such that the corresponding $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}_H$ defines a PVA structure on ${\mathcal{V}}$.

\[gfzN\] On $R_1={\mathbb{F}}[v,v^{\prime},v^{\prime\prime},\dots]$, we have the so-called *Gardner-Faddeev-Zakharov* (GFZ) PVA , given by $$\label{gfz_bracket}
\{ v_\lambda v \} = \lambda\,,$$ and the corresponding Poisson structure is $H(\partial)=\partial$. In general, for $N\geq1$, we consider the algebra of differential polynomials in $N$ variables $R_N={\mathbb{F}}[v_i^{(n)}\mid i\in\{1,\dots,N\},n\in{\mathbb{Z}}_+]$ with the generalized GFZ $\lambda$-bracket $$\{ {v_j}_\lambda {v_i} \} = s_{ij}\lambda
\,\,,\,\,\,\,
i,j\in\{1,\dots,N\}\,,$$ where $S=(s_{ij})_{i,j=1}^N$ is a symmetric matrix over ${\mathbb{F}}$. The corresponding Poisson structure is $H(\partial)=S\partial$.

\[vir-mag\] On $R_1={\mathbb{F}}[u,u^{\prime},u^{\prime\prime},\dots]$ we have the *Virasoro-Magri* PVA, given by ($c\in{\mathbb{F}}$) $$\label{virasoro_bracket}
\{u_\lambda u\}=(\partial+2\lambda)u+c\lambda^3\,,$$ with Poisson structure $H(\partial)=u^\prime+2u\partial+c\partial^3$.

Hamiltonian equations {#sub:ham_structures}
---------------------

The following proposition is immediate to check:

\[pvahamop\] Let ${\mathcal{V}}$ be a PVA. Then we have a well defined Lie algebra bracket on the quotient space ${\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$: $$\label{lambda=0}
