  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.
