On actions of reductive groups with one-parameter family

Actions of reductive groups on normal algebraic varieties with one-parameter families of spherical orbits of maximal dimension are studied under the assumption that the categorical quotient for the action is one-dimensional. As an application, the classification of the actions of the group $SL_2$ on three-dimensional normal affine varieties is completed. The ground field $K$ is assumed to be algebraically closed and of characteristic zero.