The steady and transient nonlinear rheological behaviors of dilute rod-like micellar solutions are predicted here with a particular case of the generalized Bautista-Manero-Puig (BMP) model that consists of the upper-convected Maxwell constitutive equation and a dissipative power-dependent kinetic equation, which takes into account the formation and disruption of shear-induced structures (SISs). This model has been derived using the extended irreversible thermodynamic (EIT) formalism. In steady shear, the model predicts a Newtonian region at low shear rates and a characteristic shear rate () at which shear thickening develops. In the shear thickening region, the model predicts either a reentrant zone, which is caused by multi-valued shear stresses when the data are collected with a shear stress-controlled mode or a continuous increase in the shear stress-shear rate flow curve, when the data are collected with a shear rate-controlled mode; in this region, two coexisting phases are predicted. The coexisting phases at the same shear rate in the two-phase envelope and the spinodal-like region were evaluated from the extended Maxwell equal-area criterion, which was calculated from the equal values of the two minima of the plot of the extended Gibbs free energy versus shear rate. At higher shear rates, the model predicts a transition to shear thinning, and under transient flows, an induction time and a saturation time are obtained from the predicted and the experimental data as detailed in the text. The magnitudes of both, the induction and the saturation times, diminish as the shear rate departs from , but only the induction time decreases according to a power law with shear rate. The conditions under which these rheological responses arise are derived and justified in detail with the BMP model. The model predictions are compared with experimental data of two dilute micellar solutions. The model parameters were determined from a set of independent experiments without fitting.