The Drach superintegrable systems (Q2711826)

The Drach systems are described by Hamiltonians of the form \(H=p_x p_y + U(x,y)\), which possess, in addition to \(H\), an integral of motion \(K\), cubic in the momenta. By making an ansatz on the particular form of \(K\), ten such cases for the potential \(U\) are found. The author shows that in seven of these cases the corresponding systems are superintegrable, possessing also a third intependent integral, quadratic in the momenta and separate under a point transformation. In one additional case the system may attain the separable Stäckel form by a canonical transformation which is not a point transformation. The author considers also the case of natural Hamiltonians of the form \(H=\frac{1}{2}( p_x^2 + p_y^2) + V(x,y)\), which are superintegrable in the sense that they belong to the Stäckel family and possess at the same time an integral of motion cubic in the momenta. Eleven such cases of potentials are found.