Two β-ensemble realization of β-deformed WLZZ models

We consider a two β-ensemble realization of the series of β-deformed WLZZ matrix models. We demonstrate that such a realization involves 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-ensemble model, which requires a set of identities for partition function of the one β-ensemble in the external field, and a set of identities for the β-deformed Itzykson-Zuber integral. These both sets of identities are formulated in terms of the Dunkl operators.