  We show that the factorization problem $\theta (z)=\theta_2(z)\theta_1(z)$ is
solvable in the class of Hilbert space operator-valued functions holomorphic on
some neighbourhood of $z=0$ in $\nspace{C}{N}$ and having a zero at $z=0$ (here
$\theta (z)$ has a multiple zero at $z=0$). Such a factorization problem
becomes more complicated if we demand for $\theta (z), \theta_1(z)$ and
$\theta_2(z)$ to be Agler--Schur-class functions on the polydisk
$\nspace{D}{N}$ and for the factorization identity to hold in $\nspace{D}{N}$.
In this case we reduce it to the problem on the existence of a cascade
decomposition for certain multiparametric linear system $\alpha $--a
conservative realization of $\theta (z)$, and give the criterion for its
solvability in terms of common invariant subspaces for the $N$-tuple of main
operators of $\alpha $.
