  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 $W_{1+\infty}$ algebra. We construct many such
subalgebras and explain how they look in various representations. We start from
the even simpler $w_\infty$ 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 $\beta$-deformation, an intermediate step from
$W_{1+\infty}$ 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 $W_{1+\infty}$ algebra gives rise to
KP/Toda $\tau$-functions. The hidden symmetry given by the families of
commuting Hamiltonians is in charge of the special, (skew) hypergeometric
$\tau$-functions among these.
