  We are interested in studying the variation of the Hitchin fibration in
moduli spaces of parabolic Higgs bundles, under the action of a ramified
covering. Given a degree two map $\pi$ : Y $\rightarrow$ X between compact
Riemann surfaces, we may pull back a Higgs bundle from X to Y , the lifted
Higgs bundle tends to have many apparent singularities, then we perform a
suitable birational transformation in order to eliminate them. This
correspondence preserves the Hitchin fibrations and then its restriction to a
general fiber gives a map between Abelian varieties. The aim of this paper is
to describe this map.
