  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 Painlev\'{e} type. We also show
the existence of affine Weyl group actions on solutions of such Painlev\'e type
equations, which generalizes the theory of Noumi and Yamada on affine Weyl
group symmetries of the Painlev\'{e} type equations.
