  We review the construction of Drinfeld-Sokolov type hierarchies and classical
W-algebras in a Hamiltonian symmetry reduction framework. We describe the list
of graded regular elements in the Heisenberg subalgebras of the nontwisted loop
algebra based on $gl_n$ and deal with the associated hierarchies. We exhibit an
$sl_2$ embedding for each reduction of a Kac-Moody Poisson bracket algebra to a
W-algebra of gauge invariant differential polynomials.
