  The aim of this paper is two-fold. First, we define symplectic maps between
Hitchin systems related to holomorphic bundles of different degrees. We call
these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs
bundles. They are constructed by means of the Hecke correspondence of the
underlying holomorphic bundles. SHC allows to construct B\"{a}cklund
transformations in the Hitchin systems defined over Riemann curves with marked
points. We apply the general scheme to the elliptic Calogero-Moser (CM) system
and construct SHC to an integrable $\SLN$ Euler-Arnold top (the elliptic
$\SLN$-rotator). Next, we propose a generalization of the Hitchin approach to
2d integrable theories related to the Higgs bundles of infinite rank. The main
example is an integrable two-dimensional version of the two-body elliptic CM
system. The previous construction allows to define SHC between the
two-dimensional elliptic CM system and the Landau-Lifshitz equation.
