In a recent paper (A. Mitsopoulos and M. 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 different methods, and additional ones
derived in this work can be obtained by the single algorithmic method provided
by this theorem. The results are collected in four Tables which can be used as
an updated reference of this type 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 different suitable choice of
the parameters will be able to find new integrable and superintegrable
potentials.
