\{{{\textstyle\int}}f,{{\textstyle\int}}g\}={{\textstyle\int}}\left.\{f_\lambda g\}\right|_{\lambda=0}\,.$$ Here and further, ${{\textstyle\int}}:{\mathcal{V}}\to{\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ is the canonical quotient map. Moreover, we have a well defined Lie algebra action of ${\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ on ${\mathcal{V}}$ by derivations of the commutative associative product on ${\mathcal{V}}$, commuting with $\partial$, given by $$\{{{\textstyle\int}}f,g\}=\{f_\lambda g\}|_{\lambda=0}\,.$$

\[hamsys\] Let ${\mathcal{V}}$ be a PVA. The *Hamiltonian equation* with *Hamiltonian functional* ${{\textstyle\int}}h\in{\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ is $$\label{hameq}
\frac{du}{dt}
=\{{{\textstyle\int}}h,u\}$$ An *integral of motion* for the Hamiltonian equation is an element ${{\textstyle\int}}f\in{\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ such that $\{{{\textstyle\int}}h,{{\textstyle\int}}f\}=0$ (or, equivalently, $\frac{d}{dt}{{\textstyle\int}}f=0$). Equation is called *integrable* if there exists an infinite sequence ${{\textstyle\int}}h_0={{\textstyle\int}}h,\,{{\textstyle\int}}h_1,\,{{\textstyle\int}}h_2,\dots$, of linearly independent integrals of motion in involution: $\{{{\textstyle\int}}h_m,{{\textstyle\int}}h_n\}=0$ for all $m,n\in{\mathbb{Z}}_+$. The corresponding *integrable hierarchy of Hamiltonian equations* is $$\label{eq:hierarchy}
\frac{du}{dt_n}=\{{{\textstyle\int}}h_n,u\}\,,\,\,\ n\in{\mathbb{Z}}_+\,.$$

In the special case when ${\mathcal{V}}=R_\ell$ and $H\in\operatorname{Mat}_{\ell\times\ell}{\mathcal{V}}[\partial]$ is a Poisson structure, the Lie bracket on ${\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ takes the usual form (see ): $$\{{{\textstyle\int}}f,{{\textstyle\int}}g\}_H
=\sum_{i,j\in I}\int\frac{\delta g}{\delta u_j}H_{ji}(\partial)\frac{\delta f}{\delta u_i}
\,,$$ where $\frac{\delta f}{\delta u_i}$ denotes the *variational derivative* of $f\in{\mathcal{V}}$ with respect to $u_i$, $$\label{eq:def_varder}
\frac{\delta f}{\delta u_i}=\sum_{n\in{\mathbb{Z}}_+}(-\partial)^n\frac{\partial f}{\partial u_i^{(n)}}\,,$$ (it is well defined on ${\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\partial{\mathcal{V}}}}}}$ since $\frac{\delta}{\delta u_i}\circ\partial=0$), and 