Total integrability of a Minkowski billiard in its unit ball

Let a smooth strongly convex norm on the plane define both a billiard table and the length of its chords. We prove that if the entire billiard phase annulus is foliated by continuous essential invariant curves, then the norm comes from an inner product. No finite symmetry beyond central symmetry is assumed. The proof retains the support point of each chord as an independent variable, converts length variations into alternating area variations, and averages over the two resulting phase annuli. A centro-affine boundary parameter and the Blaschke--Santaló inequality then force the unit ball to be an ellipse.