the Hamiltonian equation associated to the Hamiltonian functional ${{\textstyle\int}}h\in{\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ is, as usual, $$\frac{du_i}{dt}
=\sum_{j\in I}H_{ij}(\partial)\frac{\delta h}{\delta u_j},\ i\in I
\,.$$

Bi-PVA and the Lenard-Magri scheme of integrability {#sub:lenard_scheme}
---------------------------------------------------

\[def:compatible\] Two PVA $\lambda$-brackets $\{\cdot\,_\lambda\,\cdot\}_0$ and $\{\cdot\,_\lambda\,\cdot\}_1$ on a differential algebra ${\mathcal{V}}$ are *compatible* if any their linear combination (or, equivalently, their sum) is a PVA $\lambda$-bracket. We say in this case that ${\mathcal{V}}$ is a *bi-PVA*.

For example, the GFZ $\lambda$-bracket and the Virasoro-Magri $\lambda$-bracket are compatible.

According to the *Lenard-Magri scheme of integrability*, [@Mag78], in order to obtain an integrable hierarchy of Hamiltonian equations, one needs to find a sequence $\{{{\textstyle\int}}h_n\}_{n\in{\mathbb{Z}}_+}\subset{\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ spanning an infinite dimensional space, such that $$\label{lenardrecursion}
\{{{\textstyle\int}}h_n,u\}_1
=\{{{\textstyle\int}}h_{n+1},u\}_0
\,\,\text{ for }\,\, n\in{\mathbb{Z}}_+,\,u\in{\mathcal{V}}
\,.$$ If this is the case, then $\{{{\textstyle\int}}h_m,{{\textstyle\int}}h_n\}_\delta=0$, for all $m,n\in{\mathbb{Z}}_+$, $\delta=0,1$. Hence, we get the corresponding integrable hierarchy of Hamiltonian equations . Moreover, if $\{{{\textstyle\int}}h_0,u\}_0=0$, and $\{{{\textstyle\int}}g_n\}_{n\in{\mathbb{Z}}_+}\subset{\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ is another sequence satisfying the Lenard-Magri recursion $\{{{\textstyle\int}}g_n,u\}_1=\{{{\textstyle\int}}g_{n+1},u\}_0$, for all $n\in{\mathbb{Z}}_+$ and $u\in{\mathcal{V}}$, then the two sequences of integrals of motion are compatible: $\{{{\textstyle\int}}h_m,{{\textstyle\int}}g_n\}_\delta=0$, for all $m,n\in{\mathbb{Z}}_+$, $\delta=0,1$, [@BDSK09 Sec.2.1].

Non-local Poisson vertex algebras {#sub:non_local}
---------------------------------

For a vector space $V$, 