Linear Hamiltonians in generators of the real Jacobi group on the extended Siegel-Jacobi space and equations of motion attached

Using the energy function on the extended Siegel-Jacobi upper half space of order $n$, ${\tilde{\mathcal{X}}}_n^J$, with $n\in \mathbb{N}$, the equations of motion in the variables $(x,y,q,p,\kappa )$ attached to linear Hamiltonians in the generators of the real Jacobi group $G_n^J(\mathbb{R})$ are presented, where $x,y$ are symmetric matrices in $\mathcal{M}(n,\mathbb{R})$ and $p,q$ are real $n$-vectors. The case $n=1$ is presented separately.