Cubic first integrals of autonomous dynamical systems in E2 by an algorithmic approach

In a recent paper of Mitsopoulos and Tsamparlis [J. Geom. Phys. 170, 104383 (2021)], a general theorem is given, which provides an algorithmic method for the computation of first integrals (FIs) of autonomous dynamical systems in terms of the symmetries of the kinetic metric defined by the dynamical equations of the system. In the present work, we apply this theorem to compute the cubic FIs of autonomous conservative Newtonian dynamical systems with two degrees of freedom. We show that the known results on this topic, which have been obtained by means of various divertive methods, and the additional ones derived in this work can be obtained by the single algorithmic method provided by this theorem. The results are collected in Tables I–IV, which can be used as an updated reference for these types of integrable and superintegrable potentials. The results we find are for special values of free parameters; therefore, using the methods developed here, other researchers by a different suitable choice of the parameters will be able to find new integrable and superintegrable potentials.