  In this paper we examine the bi-Hamiltonian structure of the generalized
KdV-hierarchies. We verify that both Hamiltonian structures take the form of
Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated
system. Classical extended conformal algebras are obtained from the second
Poisson bracket. In particular, we construct the $W_n^l$ algebras, first
discussed for the case $n=3$ and $l=2$ by A. Polyakov and M. Bershadsky.
