pseudodifferential operator of order $N\geq1$.

(a) There exists exactly one monic matrix pseudodifferential operator of order one, which we denote $A^{\frac{1}{N}}(\partial)$, such that $(A^{\frac{1}{N}}(\partial))^N=A(\partial)$.

(b) There exists exactly one monic matrix pseudodifferential operator of order $-N$, which we denote $A^{-1}(\partial)$, such that $A(\partial)\circ A^{-1}(\partial)=A^{-1}(\partial)\circ A(\partial)={\mathbbm{1}}_m$.

The *symbol* of a matrix pseudodifferential operator $A(\partial)=\sum_{n\leq N}A_n\partial^n\in\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))$ is the Laurent series $A(z)=\sum_{n\leq N}A_nz^n\in\operatorname{Mat}_{m\times m}{\mathcal{A}}((z^{-1}))$, where $z$ is an indeterminate commuting with ${\mathcal{A}}$. This gives us a bijective map $\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))\longrightarrow\operatorname{Mat}_{m\times m}{\mathcal{A}}((z^{-1}))$ which is not an algebra homomorphism. For $A(\partial),B(\partial)\in\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))$, we have $$\label{eq:mult_symbol}
(A\circ B)(z)=A(z+\partial)B(z)\,.$$ Here and further, for any $n\in{\mathbb{Z}}$, we expand $(z+\partial)^n$ in non-negative powers of $\partial$ (and we let the powers of $\partial$ act to the right, on the coefficients of $B(z)$), i.e. by $(z+\partial)^n$ we mean $i_z(z+\partial)^n$.

The following lemma will be used in Section \[sec:hierarchies\].

\[lem:residui\]

(a) If $a(z),b(z)\in{\mathcal{A}}((z^{-1}))$ are formal Laurent series with coefficients in an algebra ${\mathcal{A}}$, then $\operatorname{Res}_za(z)b(z-x)=\operatorname{Res}_za(z+x)b(z)$ (where we expand $(z\pm x)^n$ in non-negative powers of $x$).

(b) If $A(\partial),B(\partial)\in{\mathcal{A}}((\partial^{-1}))$ are pseudodifferential operators over the differential algebra ${\mathcal{A}}$, then $\operatorname{Res}_zA(z)B^*(-z+\lambda)=\operatorname{Res}_zA(z+\lambda+\partial)B(z)$.

Part (a) follows from the formula of integration by parts, $\operatorname{Res}_z A(z)\partial_z B(z)=-\operatorname{Res}_zB(z)\partial_zA(z)$, and the Taylor expansion $a(z+x)=e^{x\partial_z}a(z)$. Part (b) is a special 