  We determine the homogeneous K\"ahler diffeomorphism $FC$ which expresses the
K\"ahler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as
the sum of the K\"ahler two-form on $\C^n$ and the one on the Siegel ball
$\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$
determined by a hermitian linear Hamiltonian in the generators of the Jacobi
group $G^J_n=H_n\rtimes\text{Sp}(n,\R)_{\C}$ are described by a matrix Riccati
equation on $\mc{D}_n$ and a linear first order differential equation in
$z\in\C^n$, with coefficients depending also on $W\in\mc{D}_n$. $H_n$ denotes
the $(2n+1)$-dimensional Heisenberg group. The system of linear differential
equations attached to the matrix Riccati equation is a linear Hamiltonian
system on $\mc{D}_n$. When the transform $FC:(\eta,W)\rightarrow (z,W)$ is
applied, the first order differential equation in the variable
$\eta=(\un-W\bar{W})^{-1}(z+W\bar{z})\in\C^n$ becomes decoupled from the motion
on the Siegel ball. Similar considerations are presented for the Siegel-Jacobi
upper half plane $\mc{X}^J_n=\C^n\times\mc{X}_n$, where $\mc{X}_n$ denotes the
Siegel upper half plane.
