The adiabatic auroral arc model couples a kinetic theory parallel current driven by mirror forces to horizontal ionospheric currents; the resulting equations are nonlinear. We develop some exact stationary solutions to these equations, some of them based on the Liouville equation, with both latitudinal and longitudinal spatial variations. These Liouville equation exact solutions are related to stability boundaries of low-frequency instabilities such as Kelvin-Helmholtz, as study of a simplified model shows. ¿ American Geophysical Union 1988