The set of hydrodynamic equations for the ion component of a magnetized low-pressure plasma, including the nonlinear ion drift and waves related to it, taking into account dispersion effects of order k2⊥ρ2i (k⊥is the characteristic transverse wavenumber and ρi is the ion Larmor radius), is obtained. The reduction of these equations using the standard assumptions of vortex theory is given. The problem of the integrals of motion of the simplified equations is discussed. Account is taken of the gravitational force (which models curvature of the magnetic field lines), the three-dimensionality of the perturbations (drift-Alfvén effects) and plasma rotation. It is suggested that the ion-drift hydrodynamics discussed here should be the basis for the analysis of the ion drift and the vortices related to it, as well as for the theory of decay processes with participation of the ion-drift waves.