Based on the measured transport quantities, such as electrical conductivity, we can obtain some information about electronic bands. In this paper, we consider the reverse process, starting from a microscopic model of a two-dimensional material with inversed valence bands, we calculate response functions such as DC (direct current) and optical conductivity. Under the inversion of the valence bands we consider a special class of two-dimensional semimetals valence bands that have electron-hole symmetry and are shaped like a Mexican hat. In order to study the transport properties of such materials in more detail, we defined minimal electronic effective Hamiltonian with two free parameters. Diagonalization of such a Hamiltonian, Mexican hat-shaped bands were obtained and the possibility to describe two different basic electronic phases. These are the gapped phase and the nodal surface semimetal phase. The entire paper is aimed at the fact that, based on the general form of the conductivity tensor, which we derived in the formalism the equations of motion for the non-equilibrium Landau distribution function, and which we have divided into intraband and interband excitation, we will show the unique footprint of the two mentioned electronic phases; in DC and optical response. In DC conduction, where the main excitations come from intraband processes based on the dependence of the effective electron concentration on doping, it is possible to determine the difference between the gapped phase and the nodal surface phase. This difference between phases can also be seen in the example of the density of electronic states, which for the case of a gapped phase has a root singularity, while the density of states in the nodal surface phase is a constant function of energy. The final confirmation in the identification of phases comes from interband, i.e. optical conductivity because it was shown that the optical response of the phase with the nodal surface vanishes.