on ${\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{{\mathcal{I}}}}}}$, where ${\mathcal{I}}=\langle\theta_1,\dots,\theta_m\rangle_{{\mathcal{V}}}$. However, in the special case when the constraints $\theta_1,\dots,\theta_m$ are central with respect to the first $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}_0$ we have the following result.

\[20130516:thm1\] Let ${\mathcal{V}}$ be a differential algebra, endowed with two compatible PVA $\lambda$-brackets $\{\cdot\,_\lambda\,\cdot\}_0$, $\{\cdot\,_\lambda\,\cdot\}_1$. Let $\theta_1,\dots,\theta_m\in{\mathcal{V}}$ be central elements with respect to the first $\lambda$-bracket: $\{a_\lambda\theta_i\}_0=0$ for all $i=1,\dots,m$, $a\in{\mathcal{V}}$. Let $C(\partial)=\big(C_{\alpha,\beta}(\partial)\big)_{\alpha,\beta=1}^m$ be the matrix pseudodifferential operator given by for the second $\lambda$-bracket: $C_{\alpha,\beta}(\lambda)=\{{\theta_\beta}_\lambda{\theta_\alpha}\}_1$. Suppose that the matrix $C(\partial)$ is invertible, and consider the Dirac modified PVA $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}_1^D$ given by . Then, $\{\cdot\,_\lambda\,\cdot\}_0$ and $\{\cdot\,_\lambda\,\cdot\}_1^D$ are compatible PVA $\lambda$-brackets on ${\mathcal{V}}$. Moreover, the differential algebra ideal ${\mathcal{I}}=\langle\theta_1,\dots,\theta_m\rangle_{{\mathcal{V}}}$ is a PVA ideal for both the $\lambda$-brackets $\{\cdot\,_\lambda\,\cdot\}_0$ and $\{\cdot\,_\lambda\,\cdot\}_1^D$, and we have the induced compatible PVA $\lambda$-brackets on ${\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{{\mathcal{I}}}}}}$.

Adler type pseudodifferential operators, classical W-algebras, and the Miura map {#sec:AGD}
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The Adler map for a scalar pseudodifferential operator {#subsec:adler}
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Let $L$ be a scalar pseudodifferential operator of order $\operatorname{ord}(L)=N\in{\mathbb{Z}}$, with coefficients in a differential algebra ${\mathcal{A}}$. The corresponding *Adler map* $A^{(L)}:{\mathcal{A}}((\partial^{-1}))\longrightarrow
{\mathcal{A}}((\partial^{-1}))$ is given by (cf. [@Adl79]) $$\label{adlermap}
A^{(L)}(F)=(LF)_+L-L(F L)_+
=L(FL)_--(LF)_-L
\,,$$ for any $F\in{\mathcal{A}}((\partial^{-1}))$. By the last expression in 