theory and also allows to study its matrix generalization.

Let ${\mathcal{V}}$ be a differential algebra with derivation $\partial$. Following [@Adl79], given a pseudodifferential operator $L\in{\mathcal{V}}((\partial^{-1}))$, we define the corresponding *Adler map* $A^{(L)}:\,{\mathcal{V}}((\partial^{-1}))\to{\mathcal{V}}((\partial^{-1}))$ given by , and the associated skewadjoint matrix differential operator $H^{(L)}$ with coefficients in ${\mathcal{V}}$, given by . In Theorem \[L-c\] we consider the algebra ${\mathcal{V}}_N^\infty$ of differential polynomials on the differential variables $\{u_i\}_{i\geq-N}$ and the pseudodifferential operator $L=\partial^N+u_{-N}\partial^{N-1}+u_{-N+1}\partial^{N-2}+\dots$, and we prove that, in this case, the corresponding 1-parameter family of matrices $H^{(L-c)}$ gives a bi-Poisson structure on ${\mathcal{V}}_N^\infty$. Theorem \[L-c-2\] is the analogous result for the algebra ${\mathcal{V}}_N$ of differential polynomials in finitely many variables $\{u_i\}_{i=-N}^{-1}$ and $L=\partial^N+u_{-N}\partial^{N-1}+\dots+u_{-1}$. Also in this case we get a bi-Poisson structure on ${\mathcal{V}}_N$. This is the classical ${\mathcal{W}}$-algebra associated to the Lie algebra ${\mathfrak{gl}}_N$ and its principal nilpotent element. Furthermore, performing Dirac’s reduction by the constraint $u_{-N}=0$, we obtain bi-Poisson structures on ${\mathcal{W}}_N^\infty={\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}_N^\infty}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\langle u_{-N}\rangle}}}}$ and on ${\mathcal{W}}_N={\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}_N}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\langle u_{-N}\rangle}}}}$ (the latter is the classical ${\mathcal{W}}$-algebra for ${\mathfrak{sl}}_N$).

In Section \[sec:hierarchies\] we associate to each of these bi-Poisson structures the corresponding integrable hierarchy of bi-Hamiltonian equations, using the Lenard-Magri scheme: on ${\mathcal{W}}_N^\infty$ we 