  In this article, we consider mechanical billiard systems defined with
Lagrange's integrable extension of Euler's two-center problems in the Euclidean
space, on the sphere, and in the hyperbolic space of arbitrary dimension $n \ge
3$. In the three-dimensional Euclidean space, we show that the billiard systems
with any finite combination of quadrics having two foci at the Kepler centers
are integrable. The same holds for the projections of these systems on the
three-dimensional sphere and the hyperbolic space by means of central
projection. With the same approach, we also extend these results to the
$n$-dimensional cases.
