Classical dynamical systems with the symmetry of the Kepler problem

The Hamiltonian dynamical systems of the form of H=1/2G1p2+1/2G2(xp)2+G3(xp)+U, where Gsub(j) and U are functions of r= √ x2, are investigated. The notion of the strict Kepler symmetry is introduced to single out the cases where there is the Runge-Lenz vector quadratic in the momentum. All dynamical systems with this property are found. They depend on an arbitrary function of the distance to the centrum of symmetry and two arbitrary interaction constants. The equations of motion are solved and it is shown explicitly that the orbits are closed. Cases when the strict Kepler symmetry is related to an underlying E(3) symmetry are noted. The breaking of the strict Kepler symmetry and its relation to the precession of the perihelium are discussed