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Based on the Chen--M\"oller--Sauvaget formula, we apply the theory ofintegrable systems to derive three equations for the generating series of theMasur--Veech volumes ${\rm Vol} \, \mathcal{Q}_{g,n}$ associated with theprincipal strata of the moduli spaces of quadratic differentials, and proposerefinements of the conjectural formulas given in [12,4] for the large genusasymptotics of ${\rm Vol} \, \mathcal{Q}_{g,n}$ and of the associated areaSiegel--Veech constants.

Masur--Veech volumes of quadratic differentials and their asymptotics