Abstract
We examine the billiard problem for two new integrable boundaries: two confocal parabolae with anti-parallel axes, and a confocal ellipse and hyperbolae. In each case an equation for the caustic and a second constant of motion are found geometrically. Non-focal orbits are shown to produce caustics, while focal orbits are shown to either asymptotically tend to the axis or form a closed orbit of period two or four.
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