  Discretization Program of the famous Completely Integrable Systems and
associated Linear Operators was developed in 1990s. In particular, specific
properties of the second order difference operators on the triangulated
manifolds and equilateral triangle lattices were studied in the works of
S.Novikov and I.Dynnikov since 1996. They involve factorization of operators,
the so-called Laplace Transformations, new discretization of Complex Analysis
and new discretization of $GL_n$ connections on the triangulated $n$-manifolds.
The general theory of the new type discrete $GL_n$ connections was developed.
However, the special case of $SL_n$-connections (and unimodular $SL_n^{\pm}$
connections such that $\det A=\pm 1$) was not selected properly. As we prove in
this work, it plays fundamental role (similar to magnetic field in the
continuous case) in the theory of self-adjoint discrete Schrodinger operators
for the equilateral triangle lattice in $\RR^2$. In Appendix~1 we present a
complete characterization of rank 1 unimodular $SL_n^{\pm}$ connections.
Therefore we correct a mistake made in the previous versions of our paper (we
wrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection is
equivalent to the standard Canonical Connection). Using communications of
Korepanov we completely clarify connection of classical theory of electric
chains and star-triangle with discrete Laplace transformation on the triangle
lattices
