Maria Przybylska (Institute of Physics, University of Zielona Gora, Poland)
  Integrability of homogeneous Hamiltonian systems in curved spaces.

We investigate certain natural Hamiltonian systems with two degrees of freedom. Their kinetic energy depend on coordinates and they possess appropriate forms of potentials such that these systems are homogeneous. Thanks to this property they admit particular solutions. Using these solutions we derive necessary conditions for the integrability of such systems investigating differential Galois groups of variational equations. Obtained integrability conditions give certain integrable and superintegrable systems that will be presented.