Analytical long-profile models of fluvial incision are fundamental in geomorphology, however, analogous models are missing for glaciers, despite the fact that glaciers are an important and widespread agent of physical erosion. We introduce a new analytical solution for coupled glacial-fluvial networks under steady-state climate and uplift. We then integrate this solution into a critical wedge orogen theory to explore the effect of glaciation on the balance between climate, erosion and orogen geometry. We find that the relief is closely tied to the equilibrium line altitude (ELA) and that the geometry and average uplift rate can respond significantly more sensitively to changes in climate than critical wedge orogens dominated by fluvial erosion. Using this observation, we suggest that the global lower ELA during glacial maximums over the last few millions years could trigger narrowing of critical orogens, with an associated increase in uplift rates within the active orogen core.