  Let $V$ be a symmetric space over a connected reductive Lie algebra $G$, with
Lie algebra $\mathfrak{g}$ and discriminant $\delta\in \mathbb{C}[V]$. A
fundamental object is the invariant holonomic system $\mathcal{G}
=\mathcal{D}(V)\Big/ \Bigl(\mathcal{D}(V)\mathfrak{g}+
\mathcal{D}(V)(\mathrm{Sym}\, V)^G_+ \Bigr) $ over the ring of differential
operators $\mathcal{D}(V)$. Jointly with Levasseur we have shown that there
exists a surjective radial parts map $\mathrm{rad}$ from $ \mathcal{D}(V)^G$ to
the spherical subalgebra $A_{\kappa}$ of a Cherednik algebra. When $A_{\kappa}$
is simple we show that $\mathcal{G}$ has no $\delta$-torsion submodule nor
factor module and we determine when $\mathcal{G}$ is semisimple, thereby
answering questions of Sekiguchi, respectively Levasseur-Stafford. In the
diagonal case when $V=\mathfrak{g}$, these results reduce to fundamental
theorems of Harish-Chandra and Hotta-Kashiwara. We generalise these results to
polar representations $V$ satisfying natural conditions. By twisting the radial
parts map, we obtain families of invariant holonomic systems. We introduce
shift functors between the different twists. We show that the image of the
simple summands of $\mathcal{G} $ under these functors is described by Opdam's
KZ-twist.
