The dynamics in quasihomogeneous fields (characterized by sums of homogeneous potentials) models a lot of concrete physical and astronomical situations. The two-body problem in Cartesian and standard polar coordinates, as well as in regularizing McGehee-type coordinates of both collision-blow-up kind and infinity-blow-up kind, is being tackled. The corresponding vector fields present symmetries that form isomorphic Abelian groups endowed with an