  We construct the vertex operator representation for the Affine Kac-Moody
$SL(M+K+1)$ algebra, which is relevant for the construction of the soliton
solutions of the constrained KP hierarchies. The oscillators involved in the
vertex operator construction are provided by the Heisenberg subalgebras of
$SL(M+K+1)$ realized in the unconventional gradations. The well-known limiting
cases are the homogeneous Heisenberg subalgebra of $SL(M+1)$ and the principal
Heisenberg subalgebra of ${\hat{sl}}(K+1)$. The explicit example of $M=K=1$ is
discussed in detail and the corresponding soliton solutions and tau-functions
are given.
