  In the first part of the paper, we classify linear integrable
(multi-dimensionally consistent) quad-equations on bipartite isoradial
quad-graphs in $\mathbb C$, enjoying natural symmetries and the property that
the restriction of their solutions to the black vertices satisfies a Laplace
type equation. The classification reduces to solving a functional equation.
Under certain restriction, we give a complete solution of the functional
equation, which is expressed in terms of elliptic functions. We find two real
analytic reductions, corresponding to the cases when the underlying complex
torus is of a rectangular type or of a rhombic type. The solution corresponding
to the rectangular type was previously found by Boutillier, de Tili\`ere and
Raschel. Using the multi-dimensional consistency, we construct the discrete
exponential function, which serves as a basis of solutions of the
quad-equation.
  In the second part of the paper, we focus on the integrability of discrete
linear variational problems. We consider discrete pluri-harmonic functions,
corresponding to a discrete 2-form with a quadratic dependence on the fields at
black vertices only. In an important particular case, we show that the problem
reduces to a two-field generalization of the classical star-triangle map. We
prove the integrability of this novel 3D system by showing its
multi-dimensional consistency. The Laplacians from the first part come as a
special solution of the two-field star-triangle map.
