Ancient solutions for flow by powers of the curvature in R²

We construct a new compact convex embedded ancient solution of the κ^α flow in R², α∈(12,1) that lies between two parallel lines. Using this solution we classify all convex ancient solutions of the κ^α flow in R², for α∈(23,1). Moreover, we show that any non-compact convex embedded ancient solution of the κ^α flow in R², α∈(12,1) must be a translating solution.