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We derive a quadratic recursion relation for the linear Hodge integrals of the form
$\langle \tau _{2}^{n}\lambda _{k}\rangle $
. These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.
In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the 
$\text{KP}$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of 
$\tau $-functions is given.
