A list of all integrable two-dimensional homogeneous polynomial potentials with a polynomial integral of order at most four in the momenta

We searched for integrable two-dimensional homogeneous polynomial potentials with a polynomial first integral by using the so-called direct method of searching for first integrals. We proved that there exist no polynomial first integrals which are genuinely cubic or quartic in the momenta if the degree of homogeneous polynomial potentials is greater than four.