  The $(n,m)^{\th}$ KdV hierarchy is a restriction of the KP hierarchy to a
submanifold of pseudo-differential operators in a radio form. Explicit formula
of the restricted Hamiltonian structure of KP is given which provides a new,
more constructive proof of the isomorphism between the associated
$W(n,m)$-algebra to $W_{n+m}\oplus W_m\oplus U(1)$ algebra, and the Hamiltonian
property of the $(n,m)^{\th}$ KdV hierarchy as well as its Lax-Manakov triad
representation. Similarly the Hamiltonian property for a version of modified
$n^{\th}$ KdV and the isomorphism between $W_n$-algebra to $W_l\oplus W_m\oplus
U(1)$ algebra are shown, where $l+m=n$. The role of U(1) current in both cases
is also explained.
