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

Virasoro Constraints for Drinfeld-Sokolov hierarchies and equations of Painlev\'{e} type