A rotating Stäckel potential

We study a rotating potential, which has a second integral, besides the Hamiltonian, quadratic in the momenta. This can be expressed as a Stäckel potential in elliptic coordinates, but it is nonseparable, unless its rotation is zero, We find the canonical momenta corresponding to the elliptic coordinates and the forms of the Hamiltonian and of the new integral in elliptic coordinates and momenta. The forms of the orbits are found numerically and analytically. In the nonrotating case the orbits fill either an ellipse around both foci, or a region around one focus limited by an ellipse and a hyperbola. In the rotating case the orbits are tubes, either around both foci or around only one focus. Of particular interest are the periodic orbits, which we derive analytically