  We study the conformal properties of the multi-constraint KP hierarchy and
its nonstandard partner by covariantizing their corresponding Lax operators.
The associated second Hamiltonian structures turn out to be nonlocal extension
of $W_n$ algebra by some integer or half-integer spin fields depending on the
order of the Lax operators. In particular, we show that the complicated second
Hamiltonian structure of the nonstandard multi-constraint KP hierarchy can be
simplified by factorizing its Lax operator to multiplication form. We then
diagonalize this simplified Poisson matrix and obtain the free-field
realizations of its associated nonlocal algebras.
