  We classify the surfaces translating under the flows by sub-affine-critical
powers of the Gauss curvature. This, in particular, lists all translating
solitons possibly model Type II singularities for convex closed solutions in
all positive powers. The surfaces are entire graphs, and therefore our result
corresponds to the Liouville theorem for the degenerate Monge--Amp\`ere
equations $\det D^2 u=(1+|Du|^2)^{2-\frac{1}{2\alpha}}$ on $\mathbb{R}^2$ in
the range $0<\alpha <1/4$. The result also reveals that the moduli spaces of
solutions are homeomorphic to either Euclidean spaces or cylinders.
