Much as non-Newtonian fluids exhibit secondary flows in elliptic pipes, rarefied gases exhibit secondary heat flows between elliptical cylinders. The phenomenon may be called non-Fourierian, because it is not covered by Fourier's law of heat conduction. The effect is demonstrated in the paper by exploiting the 13-moment theory of gases. Apart from secondary heat flows the theory predicts shear stresses balanced by gradients of the heat flux, and the need for a distinction of the kinetic and the thermodynamic temperature. (c) 2008 Elsevier B.V. All rights reserved.