  Motivated by the search for new backgrounds with integrable string theories,
we classify the D-brane geometries leading to integrable geodesics. Our
analysis demonstrates that the Hamilton-Jacobi equation for massless geodesics
can only separate in elliptic or spherical coordinates, and all known
integrable backgrounds are covered by this separation. In particular, we
identify the standard parameterization of AdS_p X S^q with elliptic coordinates
on a flat base. We also find new geometries admitting separation of the
Hamilton-Jacobi equation in the elliptic coordinates. Since separability of
this equation is a necessary condition for integrability of strings, our
analysis gives severe restrictions on the potential candidates for integrable
string theories.
