Doubly Exotic Nth-Order Superintegrable Classical Systems Separating in Cartesian Coordinates

Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space E₂ are explored. The study is restricted to Hamiltonians allowing separation of variables V (x, y) = V₁(x) + V₂(y) in Cartesian coordinates. In particular, the Hamiltonian H admits a polynomial integral of order N > 2. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of the formalism are non-linear compatibility conditions and the algebra of the integrals of motion. The case N = 5, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case N > 2 and a formulation of inverse problem in superintegrability are briefly discussed as well.