Two β-ensemble realization of β-deformed WLZZ models

We consider a two β β -ensemble realization of the series of β β -deformed WLZZ matrix models. We demonstrate that such a realization involves β β -deformed Harish-Chandra–Itzykson–Zuber integrals, one of them providing a coupling to the external field. We also construct Ward identities in the corresponding two β β -ensemble model, which requires a set of identities for partition function of the one β β -ensemble in the external field, and a set of identities for the β β -deformed Itzykson-Zuber integral. These both sets of identities are formulated in terms of the Dunkl operators.