It was demonstrated recently that the $W_{1+\infty }$ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized $\tilde{W}$ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the $\tilde{W}$ algebras studied earlier. We suggest a definition of the generalized $\tilde{W}$ algebra as differential operators in variables $p_k$ basing on the matrix realization of the $W_{1+\infty }$ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the $W_{1+\infty }$ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with $\tilde{W}$ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric τ-functions corresponding to the completed cycles.