The highly agitated flow of grains down an inclined chute is modelled using a kinetic theory for inelastic collisions. Solutions corresponding to steady, fully developed flows are obtained by solving numerically a nonlinear system of ordinary differential equations using a highly accurate pseudospectral method based on mapped Chebyshev polynomials. The solutions are characterized by introducing macroscopic, depth-integrated variables representing the mass flux of flowing material per unit width, its centre-of-mass and the mass supported within the flowing layer, and the influence of the controlling parameters on these solutions is investigated. It is shown that, in certain regions of parameter space, multiple steady solutions can be found for a specified mass flux of material. An asymptotic analysis of the governing equations, appropriate to highly agitated flows, is also developed and these results aid in the demarcation of domains in parameter space where steady solutions can be obtained.