Derivation of an equation of phenomenological symmetry for some three-dimensional geometries

The main problems of the theory of phenomenologically symmetric (PS) geometries, i.e., geometries of maximum mobility, are their complete classification, the establishing of the fact of existence of their group symmetry, and finding of an equation of the phenomenological symmetry for each of them. A complete classification of three-dimensional PS geometries has been already built. Their PS, i.e., the existence of a functional relation between the values of the metric function for all pairs of five points follows from the rank of the corresponding functional matrix. However, not for all such geometries an equation which expresses the PS is known in the explicit form. The paper describes methods of finding the equations of PS which were applied to some three-dimensional geometries. For each of them we give groups of motions that define the group symmetry of degree six.