  We completely characterize point--line configurations with $\Theta(n^{4/3})$
incidences when the point set is a section of the integer lattice. This can be
seen as the main special case of the structural Szemer\'edi-Trotter problem. We
also derive a partial characterization for several generalizations: (i) We rule
out the concurrent lines case when the point set is a Cartesian product of an
arithmetic progression and an arbitrary set. (ii) We study the case of a
Cartesian product where one or both sets are generalized arithmetic
progression. Our proofs rely on deriving properties of multiplicative energies.
