  Posing Kepler's problem of motion around a fixed "sun" requires the geometric
mechanician to choose a metric and a Laplacian. The metric provides the kinetic
energy. The fundamental solution to the Laplacian (with delta source at the
"sun") provides the potential energy. Posing Kepler's three laws (with input
from Galileo) requires symmetry conditions. The metric space must be
homogeneous, isotropic, and admit dilations. Any Riemannian manifold enjoying
these three symmetry properties is Euclidean. So if we want a semblance of
Kepler's three laws to hold but also want to leave the Euclidean realm, we are
forced out of the realm of Riemannian geometries. The Heisenberg group (a
subRiemannian geometry) and lattices provide the simplest examples of metric
spaces enjoying a semblance of all three of the Keplerian symmetries. We report
success in posing, and solving, the Kepler problem on the Heisenberg group. We
report failures in posing the Kepler problem on the rank two lattice and
partial success in solving the problem on the integers. We pose a number of
questions.
