  In this paper, we use the projective dynamical approach to integrable
mechanical billiards as in [Zhao, 2021] to establish the integrability of
natural mechanical billiards with the Lagrange problem, which is the
superposition of two Kepler problems and a Hooke problem, with the Hooke center
at the middle of the Kepler centers, as the underlying mechanical systems, and
with any combinations of confocal conic sections with foci at the Kepler
centers as the reflection wall, in the plane, on the sphere, and in the
hyperbolic plane. This covers many previously known integrable mechanical
billiards, especially the integrable Hooke, Kepler and two-center billiards in
the plane, as has been investigated in [Takeuchi-Zhao, 2021], as subcases. The
approach of [Takeuchi-Zhao, 2021] based on conformal correspondence has been
also applied to integrable Kepler billiards in the hyperbolic plane to
illustrate their equivalence with the corresponding integrable Hooke billiards
on the hemisphere and in the hyperbolic plane as well.
