  We construct a new compact convex embedded ancient solution of the
$\kappa^\alpha$ flow in $\mathbb R^2$, $\alpha\in(\frac12,1)$ that lies between
two parallel lines. Using this solution we classify all convex ancient
solutions of the $\kappa^\alpha$ flow in $\mathbb R^2$, for
$\alpha\in(\frac23,1)$. Moreover, we show that any non-compact convex embedded
ancient solution of the $\kappa^\alpha$ flow in $\mathbb R^2$,
$\alpha\in(\frac12,1)$ must be a translating solution.
