Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces

We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on E-2 as well as their equivalents on S-2 and H-2 for which these necessary conditions are also sufficient. We show explicit forms of the additional first integrals which can always be chosen as a polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta.