  In this short review we compare constructions of 2d integrable models by
means of two gauge field theories. The first one is the 4d Chern-Simons (4d-CS)
theory proposed by Costello and Yamazaki. The second one is the 2d
generalization of the Hitchin integrable systems constructed by means the
Affine Higgs bundles (AHB). We illustrate this approach by considering 1+1
field versions of elliptic integrable systems including the Calogero-Moser
field theory, the Landau-Lifshitz model and the field theory generalization of
the elliptic Gaudin model.
