  We give a beautiful explicit example of a convex plane curve such that the
outer billiard has a given finite number of invariant curves. Moreover, the
dynamics on these curves is a standard shift. This example can be considered as
an outer analog of the so-called Gutkin billiard tables. We test total
integrability of these billiards, in the region between the two invariant
curves. Next, we provide computer simulations on the dynamics in this region.
At first glance, the dynamics looks regular but by magnifying the picture we
see components of chaotic behavior near the hyperbolic periodic orbits. We
believe this is a useful geometric example for coexistence of regular and
chaotic behavior of twist maps.
