Maciejewski A.J., Przybylska M., Tsiganov A.V. On certain classes of integrable and superintegrable homogeneous potentials, talk Workshop GDEq Integr., 11-15 Oct. 2010, Hradec nad Moravici, Czech Rep. (abstr.)

We study the integrability of two-dimensional natural Hamiltonian systems with indefinite form of kinetic energy and homogeneous potentials of degree ${\displaystyle k}$ of the form ${\displaystyle H=2p_{1}p_{2}+q_{1}^{l}q_{2}^{k-l}}$, where ${\displaystyle k}$, ${\displaystyle l}$ are rational numbers. We show results of application of the direct method for search of first integrals. Three classes of integrable potentials are presented for that separations of variables are made and superintegrable cases are specified. In this aim some solvability results for hypergeometric equations are applied.