the most powerful approaches to classical ${\mathcal{W}}$-algebras is via the Drinfeld-Sokolov Hamiltonian reduction, [@DS85], associated to any simple Lie algebra and its principal nilpotent element. Later on this theory was extended to more general nilpotent elements, and it was put by [@DSKV13a] in the framework of Poisson vertex algebras (PVA).

However the most important classical ${\mathcal{W}}$-algebras, those associated to ${\mathfrak{sl}}_N$ and its principal nilpotent element, have appeared prior to [@DS85] together with their deep connection to integrable systems. They were identified with the so-called second Poisson structure of the $N$-th (or generalized) KdV equations, the KdV equation corresponding to the case of $N=2$. The first Poisson structure of the KdV equation was found by [@Gar71] and [@ZF71]. Later, it was shown by [@Mag78], that there exists not only a Poisson structure but a one-parameter family of compatible Poisson structures for the KdV equation. In this case we say that we have a bi-Poisson structure. The fact that the same equation could be written in two different Hamiltonian forms has become fundamental in proving integrability of such equations using a recurrence procedure, nowadays called the Lenard-Magri scheme of integrability.

The generalization to the $N$-th KdV hierarchy was suggested by [@GD76] who 