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Virasoro Constraints for Drinfeld-Sokolov hierarchies and equations of Painlevé type

We construct a tau cover of the generalized Drinfeld-Sokolov hierarchy associated to an arbitrary affine Kac-Moody algebra with gradations $\mathrm{s}\le\mathds{1}$ and derive its Virasoro symmetries. By imposing the Virasoro constraints we obtain solutions of the Drinfeld-Sokolov hierarchy of Witten-Kontsevich and of Brezin-Gross-Witten types, and of those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on solutions of such Painlevé type equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé type equations.