  The structure of the electronic nonlinear optical conductivity is elucidated
in a detailed study of the time-reversal symmetric two-band model. The
nonlinear conductivity is decomposed as a sum of contributions related with
different regions of the First Brillouin Zone, defined by single or multiphoton
resonances. All contributions are written in terms of the same integrals, which
contain all information specific to the particular model under study. In this
way, ready-to-use formulas are provided that reduce the often tedious
calculations of the second and third order optical conductivity to the
evaluation of a small set of similar integrals. In the scenario where charge
carriers are present prior to optical excitation, Fermi surface contributions
must also be considered and are shown to have an universal frequency
dependence, tunable by doping. General characteristics are made evident in this
type of resonance-based analysis: the existence of step functions that
determine the chemical potential dependence of electron-hole symmetric
insulators; the determination of the imaginary part by Hilbert transforms,
simpler than those of the nonlinear Kramers-Kr\"{o}nig relations; the absence
of Drude peaks in the diagonal elements of the second order conductivity, among
others. As examples, analytical expressions are derived for the nonlinear
conductivities of some simple systems: a very basic model of direct gap
semiconductors and the Dirac fermions of monolayer graphene.
