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A system of equations is introduced and discussed that describe the nonlinear dynamics of magnetic perturbations in a magnetized, high-temperature plasma. Diamagnetism, ion gyroradii effects, and finite electron mass are taken into account. These equations govern Alfvén as well as electrostatic waves and vortices and describe the nonlinear evolution of reconnecting modes. Electrons are treated in a fluid model. The equation for the ion response is new and is a nonlinear generalization to all orders in the thermal ion gyroradius of the nonlinear fluid model. This system of equations conserves two fluxes that are different from, but related to, the magnetic flux. Two-dimensional equilibrium solutions in the form of stationary propagating magnetic structures are obtained with the methods introduced in the theory of vector nonlinearities in electrostatic drift vortices. In the noncollisional regimes of interest the inertia of the electrons resolves the singularity in the current density that tends to develop at magnetic separatrices. The positions of the X points of the conserved fluxes are mirror symmetric and at a distance of the order of the electron skin depth from the resonant surface. The set of equations admits an energy integral and can be cast in noncanonical Hamiltonian form. The role of the Casimir invariants, that are functions of the conserved fluxes, is investigated and the connection with ‘‘reduced magnetohydrodynamics’’ is emphasized.