The existence of solutions of the non-sliding shallow-ice-sheet equation on a flat horizontal bed with a mass balance linearly depending on altitude is proven for fixed margins. Free-margin solutions for the same mass balance do not exist. Fixed-margin solutions show unbounded shear stress and nonzero mass flux at the margin. Steady-state solutions with realistic margins, vanishing ice flux and vanishing shear stress are found numerically for ice sheets with Weertman-type sliding.