New Coordinate Systems and Potentials on a Plane Providing Unusual   Intergral of Motion
  We discuss a new class of coordinate systems for a plane, which provide an analytical representation of arbitrary straightline, and then define the form of potential on the plane, under which the equations of motion of a mass point are solvable. In both cases we use the Hamiltonian approach, reducing the problem to the Hamilton-Jacobi equation. The coordinate systems in question are defined in a somewhat unusual way, and, correspondingly, the potentials we obtain, also possess some unusual properties. Since the equations of motion are solvable within this potential, the second integral of motion is defined too, and, like the coordinate system and potential, it possesses some nonstandard properties. 
Z. Y. Turakulov