Criterion of transverse stability for ion traps with transverse motion integrable in elliptic coordinates

Electrostatic traps with ideal spatiotemporal focusing of ions in one direction of their motion and the motion in transverse directions integrable in the elliptic coordinates are considered. Conditions for the confinement of ions in the trapping field are determined. The finite character of ion motion under the proposed confinement conditions is independently checked by numerical integration of the Lagrange equations.