case of (a), when $x=\lambda+\partial$, acting on the coefficients of $B$.

Poisson vertex algebras {#sub:pva}
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\[def:lambda\] A $\lambda$-*bracket* on a differential algebra ${\mathcal{V}}$ is an ${\mathbb{F}}$-linear map $\{\cdot\,_\lambda\,\cdot\}:\,{\mathcal{V}}\otimes{\mathcal{V}}\to{\mathcal{V}}[\lambda]$ satisfying ($f,g,h\in{\mathcal{V}}$)

(i) *sesquilinearity*: $\{\partial f_\lambda g\}=-\lambda\{f_\lambda g\}$, $\{f_\lambda\partial g\}=(\lambda+\partial)\{f_\lambda g\}$,

(ii) *Leibniz rules*: $\{f_\lambda gh\}=\{f_\lambda g\}h+\{f_\lambda h\}g$, $\{fh_\lambda g\}=\{f_{\lambda+\partial}g\}_{\rightarrow}\!h+\{h_{\lambda+\partial}g\}_{\rightarrow}\!f$.

Here and further we use the following notation: if $\{f_\lambda g\}=\sum_{n\in{\mathbb{Z}}_+}\lambda^n c_n$, then $\{f_{\lambda+\partial}g\}_{\rightarrow}h=\sum_{n\in{\mathbb{Z}}_+}c_n(\lambda+\partial)^nh$.

\[def:pva\] A *Poisson vertex algebra* (PVA) is a differential algebra ${\mathcal{V}}$ endowed with a $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}$ satisfying ($f,g,h\in{\mathcal{V}}$)

(iii) *skew-symmetry*: $\{g_\lambda f\}=-\{f_{-\lambda-\partial}g\}$,

(iv) *Jacobi identity*: $\{f_\lambda\{g_\mu h\}\}-\{g_\mu\{f_\lambda h\}\}=\{\{f_\lambda g\}_{\lambda+\mu}h\}$.

In the skew-symmetry we use the following notation: if $\{f_\lambda g\}=\sum_{n\in{\mathbb{Z}}_+}\lambda^n c_n$, then $\{f_{-\lambda-\partial}g\}=\sum_{n\in{\mathbb{Z}}_+}(-\lambda-\partial)^nc_n$.

\[def:CFTtype\] Let ${\mathcal{V}}$ be a PVA. A *Virasoro element* $T\in{\mathcal{V}}$ with *central charge* $c\in{\mathbb{F}}$ is such that $\{T_\lambda T\}=(2\lambda+\partial)T+c\lambda^3$, a $T$-*eigenvector* $a\in{\mathcal{V}}$ of *conformal weight* $\Delta_a\in{\mathbb{F}}$ is such that $\{T_\lambda a\}=(\Delta_a\lambda+\partial)a+O(\lambda^2)$, and a *primary element* $a\in{\mathcal{V}}$ of conformal weight $\Delta_a$ is such that $\{T_\lambda a\}=(\Delta_a\lambda+\partial)a$. By definition, a *PVA of CFT* (conformal field theory) *type* is generated, as a differential algebra, by a Virasoro element and a finite number of primary elements.

Let ${\mathcal{I}}\subset{\mathcal{V}}$ be a differential ideal. We say that ${\mathcal{I}}$ is a *PVA ideal* if $\{{\mathcal{I}}_\lambda{\mathcal{V}}\}\subset{\mathcal{I}}[\lambda]$ (by skew-symmetry and the fact that 