  A complex integrable system determines a family of complex tori over a
Zariski-open and dense subset in its base. This family in turn yields an
integral variation of Hodge structures of weight $\pm 1$. In this paper, we
study the converse of this procedure. Starting from an integral variation of
Hodge structures of weight $\pm 1$, we give a criterion for when its associated
family of complex tori carries a Lagrangian structure, i.e. for when it can be
given the structure of an integrable system. This sheaf-theoretic approach to
(the smooth parts of) complex integrable systems enables us to apply powerful
tools from Hodge and sheaf theory to study complex integrable systems. We
exemplify the usefulness of this viewpoint by proving that the degree zero
component of the Hitchin system for any simple adjoint or simply-connected
complex Lie group $G$ is isomorphic to a non-compact Calabi-Yau integrable
system over a Zariski-open and dense subset in the corresponding Hitchin base.
In particular, we recover previously known results for the case where $G$ has a
Dynkin diagram of type ADE and extend them to the remaining Dynkin types
$\mathrm{B}_k$, $\mathrm{C}_k$, $\mathrm{F}_4$ and $\mathrm{G}_2$.
