  In 2018, Vladimir Fock and the author introduced a geometric structure on
surfaces, called higher complex structure, whose moduli space shares several
properties with Hitchin's component. In this paper, we establish various links
between flat connections and higher complex structures.
  In particular, we study a certain class of connections on a bundle equipped
with a line subbundle $L$, which we call $L$-parabolic. The curvature of these
connections is of rank 1. A certain family of $L$-parabolic connections is
parametrized by the data of a higher complex structure and a cotangent
variation. The family of connections being flat implies the compatibility
condition of the cotangent variation. Higher diffeomorphisms are realized by
the gauge transformation induced by changing $L$. Constructing flat families of
connections of this kind is linked to Toda integrable systems.
