Note on integrability of certain homogeneous Hamiltonian systems in 2D   constant curvature spaces
  We formulate the necessary conditions for the integrability of a certain family of Hamiltonian systems defined in the constant curvature two-dimensional spaces. Proposed form of potential can be considered as a counterpart of a homogeneous potential in flat spaces. Thanks to this property Hamilton equations admit, in a general case, a particular solution. Using this solution we derive necessary integrability conditions investigating differential Galois group of variational equations. 
Andrzej J. Maciejewski, Wojciech Szumi\'nski and Maria Przybylska
nlin.SI math-ph math.MP nlin.CD