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Let $V$ be a symmetric space over a connected reductive Lie algebra $G$, withLie algebra $\mathfrak{g}$ and discriminant $\delta\in \mathbb{C}[V]$. Afundamental object is the \emph{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 differentialoperators $\mathcal{D}(V)$. Jointly with Levasseur we have shown that thereexists a surjective radial parts map $\mathrm{rad}$ from $ \mathcal{D}(V)^G$ tothe spherical subalgebra $A_{\kappa}$ of a Cherednik algebra. When $A_{\kappa}$is simple we show that $\mathcal{G}$ has no $\delta$-torsion submodule norfactor module and we determine when $\mathcal{G}$ is semisimple, therebyanswering questions of Sekiguchi, respectively Levasseur-Stafford. In thediagonal case when $V=\mathfrak{g}$, these results reduce to fundamentaltheorems of Harish-Chandra and Hotta-Kashiwara.  We generalise these results to polar representations $V$ satisfying naturalconditions.  By twisting the radial parts map, we obtain families of invariant holonomicsystems. We introduce shift functors between the different twists. We show thatthe image of the simple summands of $\mathcal{G} $ under these functors isdescribed by Opdam's KZ-twist.

Invariant holonomic systems on symmetric spaces and other polar representations