We examine kinematic dynamo action in an electrically conducting spherical body of fluid with an overlying shell of differential rotation. It is thought that such flow types could explain the nearly axisymmetric magnetic field of Saturn. Although we find that an outer layer of differential rotation can 'axisymmetrize' the exterior magnetic field, surprisingly, sometimes it does not, giving magnetic fields that have no axisymmetric ingredients. ¿ 2000 American Geophysical Union