calculus which will be used throughout the paper, cf. [@Kac96].

Given a vector space ${\mathcal{A}}$, an ${\mathcal{A}}$-valued formal distribution in $z$ is a series of the form $a(z)=\sum_{n\in{\mathbb{Z}}}a_nz^n$, where $a_n\in{\mathcal{A}}$. They form a vector space denoted by ${\mathcal{A}}[[z,z^{-1}]]$. An ${\mathcal{A}}$-valued formal distribution in two variables $z$ and $w$ is a series of the form $a(z,w)=\sum_{m,n\in{\mathbb{Z}}}a_{mn}z^mw^n$, where $a_{mm}\in{\mathcal{A}}$. They form a vector space denoted by ${\mathcal{A}}[[z,z^{-1},w,w^{-1}]]$.

The $\delta$-*function* is, by definition, the ${\mathbb{F}}$-valued formal distribution $$\delta(z-w)
=\sum_{n\in{\mathbb{Z}}}z^{-n-1}w^n\in{\mathbb{F}}[[z,z^{-1},w,w^{-1}]]\,.$$ For every $a(z)\in{\mathcal{A}}[[z,z^{-1}]]$ we have $$\label{deltaprop}
a(z)\delta(z-w)
=a(w)\delta(z-w)\,.$$ In particular, $\operatorname{Res}_za(z)\delta(z-w)=a(w)$, where $\operatorname{Res}_z$ denotes the coefficient of $z^{-1}$.

We denote by $i_z$ the power series expansion for large $|z|$. For example, $$i_z(z-w)^{-1}=\sum_{k\in{\mathbb{Z}}_+}z^{-k-1}w^k
\,.$$ Using this notation, the $\delta$-function can be rewritten as follows $$\label{delta}
\delta(z-w)=i_{z}(z-w)^{-1}-i_{w}(z-w)^{-1}\,.$$ For $a(z)=\sum_{n\in{\mathbb{Z}}}a_nz^n\in{\mathcal{A}}[[z,z^{-1}]]$, we denote $a(z)_+=\sum_{n\in{\mathbb{Z}}_+}a_nz^n$ and $a(z)_-=\sum_{n<0}a_nz^n$. It is easy to check that $$\label{20130927:cor1}
\operatorname{Res}_za(z)i_{z}(z-w)^{-1}
=a(w)_+
\,.$$

The algebra of matrix pseudodifferential operators {#sub:pseudo}
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By a *differential algebra* we mean a unital commutative associative algebra ${\mathcal{A}}$ over a field ${\mathbb{F}}$ of characteristic $0$, with a derivation $\partial$.

The algebra $\operatorname{Mat}_{m\times m}{\mathcal{A}}$ of matrices with coefficients in ${\mathcal{A}}$ is a unital associative algebra, with the obvious action of $\partial$ as a derivation. We consider the algebra $\operatorname{Mat}_{m\times m}{\mathcal{A}}((\partial^{-1}))$ of 