Commutative subalgebras of the Affine Yangian $Y(\hat{gl}_1)$ and matrix models

We explain that the study of the connection between W-representations of matrix models and their superintegrability naturally leads to commutative subalgebras of the W_\infty algebra. These subalgebras correspond to new integrable systems generalizing the Calogero family. We construct many such subalgebras and explain how they look in various representations. We explain that some of the subalgebras survive the β-deformation, an intermediate step from W_\infty to the affine Yangian. This deformation is natural on the side of many-body systems, where it actually introduces the coupling. The hidden symmetry given by the families of commuting Hamiltonians is in charge of the special, (skew) hypergeometric τ-functions.