On certain integrable and superintegrable weight-homogeneous Hamiltonian systems

In this paper we deal with integrability and superintegrability of a certain class of 2-dimensional, weight-homogeneous generalizations of natural Hamiltonian systems in polar coordinates. For this purpose, we use a version of the so-called direct method of searching for polynomial first integrals of such systems. As results we found certain integrable as well as superintegrable cases.