Polynomially superintegrable Hamiltonians separating in Cartesian coordinates

Abstract The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The computation of the compatibility equations requires a general formula for the coefficients, which in turn must depend on the potential to be solved for. This is in general a nonlinear problem and quite difficult. Thus, research has focused on dividing the classes of potential into standard and exotic ones so that a number of parameters may be set to zero and the coefficients may be obtained in a simpler setting. We have developed a new method in both the classical and quantum settings that allows a formula for the coefficients of the integral to be obtained without recourse to this division for Cartesian-separable Hamiltonians. The expressions we obtain are in general non-polynomial in the momenta whose fractional terms can be arbitrarily set to zero. These conditions are equivalent to the compatibility equations, but the only unknowns in addition to the potential are constant parameters. We also classify all the fourth-order standard superintegrable Hamiltonians.