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Let $r\geq 2$ be an integer. The generalized BGW tau-function for theGelfand--Dickey hierarchy of $(r-1)$ dependent variables (aka the $r$-reducedKP hierarchy) is defined as a particular tau-function that depends on $(r-1)$constant parameters $d_1,\dots,d_{r-1}$. In this paper we show that thistau-function satisfies a family of linear equations, called the $W$-constraintsof the second kind. The operators giving rise to the linear equations alsodepend on $(r-1)$ constant parameters. We show that there is a one-to-onecorrespondence between the two sets of parameters.

Gelfand--Dickey hierarchy, generalized BGW tau-function, and $W$-constraints