Hitchin maps and parabolic Hitchin maps on the moduli spaces of Hitchin sheaves on nodal curves

We study the Hitchin maps on the moduli spaces of Hitchin sheaves and parabolic Hitchin sheaves on a nodal integral curve $Y$. We study their fibres, the BNR correspondences and the relation of the restriction of the Hitchin map with very stable bundles. As an application, in the "coprime" case we prove that the parabolic Poincaré sheaf on $U_{par}(n, ξ)\times Y$ is parabolic stable where $U_{par}(n, ξ)$ is a fixed determinant moduli space of parabolic sheaves on $Y$.