Numerical non-integrability of Hexagonal string billiard

We consider a remarkable $C^2$-smooth billiard table introduced by Hans L.Fetter. It is obtained by the string construction from a regular hexagon for a special value of the length of the string. It was suggested as a possible counter-example to the Birkhoff-Poritsky conjecture. In this paper, we investigate numerically the behavior of this billiard and find chaotic regions near hyperbolic periodic orbits. They are very small since the billiard table is nearly circular.