obtain the KP hierarchy, while on ${\mathcal{W}}_N$ we obtain the $N$-th KdV hierarchy.

Furthermore, in Section \[sub:extra2\] we use the Adler map to define natural PVA homomorphisms ${\mathcal{V}}_{M+N}^\infty\to{\mathcal{V}}_M^\infty\otimes{\mathcal{V}}_N^\infty$ and ${\mathcal{V}}_{M+N}\to{\mathcal{V}}_M\otimes{\mathcal{V}}_N$, which are a generalization of the usual “Miura map”, [@Miu68]. This allows us to give another proof of the Theorem of [@KW81].

A major advantage of our approach is that all the above constructions and proofs extend in a straightforward way to the case when the variables $u_i$ are replaced by $m\times m$ matrices $U_i$. This is done in Section \[sec:matrixAGD\], where, as a result, we construct the matrix KP and $N$-th KdV bi-Hamiltonian equations. The main difference with the scalar case is that, in the matrix case, Dirac reduction (developed by [@DSKV13c]) by the constraint $U_{-N}=0$ leads to a non-local bi-Poisson structure. These non-local Poisson structures have been discovered by [@Bil95] and [@OS98]. We then need to use the theory of non-local Poisson structures introduced by [@DSK13], and the machinery of rational matrix pseudodifferential operators developed by [@CDSK13b], to prove integrability of the matrix $N$-th KdV hierarchy.

Preliminaries {#sec:back}
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Some simple facts on formal distribution calculus {#sub:formal}
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We briefly review here some basic facts on formal distribution 