  We study the existence and uniqueness of the Kadomtsev-Petviashvili (KP)
hierarchy solutions in the algebra of $\F Cl(S^1,\K^n)$ of formal classical
pseudo-differential operators. The classical algebra $\Psi DO(S^1,\K^n)$ where
the KP hierarchy is well-known appears as a subalgebra of $\F Cl(S^1,\K^n).$ We
investigate algebraic properties of $\F Cl(S^1,\K^n)$ such as splittings,
r-matrices, extension of the Gelfand-Dickii bracket, almost complex structures.
Then, we prove the existence and uniqueness of the KP hierarchy solutions in
$\F Cl(S^1,\K^n)$ with respect to extended classes of initial values. Finally,
we extend this KP hierarchy to complex order formal pseudo-differential
operators and we describe their Hamiltonian structures similarly to previously
known formal case.
