  Recently, a new generalized family of infinite-dimensional $ \widetilde{W} $
algebras, each associated with a particular element of a commutative subalgebra
of the $ W_{1+\infty} $ algebra, was described. This paper provides a
comprehensive account of the aforementioned association, accompanied by the
requisite proofs and illustrative examples. This approach allows a derivation
of Ward identities for selected WLZZ matrix models and the expansion of
corresponding $ W $-operators in terms of an infinite set of variables $ p_k $.
