Commutative families in W ∞ , integrable many-body systems and hypergeometric τ -functions

bstract We explain that the set of new integrable systems, generalizing the Calogero family and implied by the study of WLZZ models, which was described in arXiv:2303.05273 , is only the tip of the iceberg. We provide its wide generalization and explain that it is related to commutative subalgebras (Hamiltonians) of the W 1+ ∞ algebra. We construct many such subalgebras and explain how they look in various representations. We start from the even simpler w ∞ contraction, then proceed to the one-body representation in terms of differential operators on a circle, further generalizing to matrices and in their eigenvalues, in finally to the bosonic representation in terms of time-variables. Moreover, we explain that some of the subalgebras survive the β -deformation, an intermediate step from W 1+ ∞ to the affine Yangian. The very explicit formulas for the corresponding Hamiltonians in these cases are provided. Integrable many-body systems generalizing the rational Calogero model arise in the representation in terms of eigenvalues. Each element of W 1+ ∞ algebra gives rise to KP/Toda τ -functions. The hidden symmetry given by the families of commuting Hamiltonians is in charge of the special, (skew) hypergeometric τ -functions among these.