On an extension of the generalized BGW tau-function

For an arbitrary solution to the Burgers--KdV hierarchy, we define the tau-tuple $(\tau_1,\tau_2)$ of the solution. We show that the product $\tau_1\tau_2$ admits Buryak's residue formula. Therefore, according to Alexandrov's theorem, $\tau_1\tau_2$ is a tau-function of the KP hierarchy. We then derive a formula for the affine coordinates for the point of the Sato Grassmannian corresponding to the tau-function $\tau_1\tau_2$ explicitly in terms of those for $\tau_1$. Applications to the analogous open extension of the generalized BGW tau-function and to the open partition function are given.