  Very recently Ben Andrews and Haizhong Li showed that every embedded cmc
torus in the three dimensional sphere is axially symmetric. There is a
two-parametric family of axially symmetric cmc surfaces; more precisely, for
every real number H and every C > 2 (H+\sqrt{1+H^2}) there is an axially
symmetry surface \Sigma_{H,C} with mean curvature H. In 2010, Perdomo showed
that for every H between cot(\pi/m) and (m^2-2)/(2(m^2-1)^1/2), there exists an
embedded axially symmetric example with non constant principal curvatures that
is invariant under the ciclic group Z_m. Andrews and Li, showed that these
examples are the only non-isoparametric embedded examples in the family when
H>0. In this paper we study those examples in the family with H<0. We prove
that there are no embedded examples in the family when H<0 and we also prove
that for every integer m>2 there is a properly immersed example in this family
that contains a great circle and is invariant under the ciclic group Z_m. We
will say that these examples contain the axis of symmetry. Finally we show that
every non-isoparametric surface \Sigma_{H,C} is either properly immersed
invariant under the ciclic group Z_m for some integer m>1 or it is dense in the
region bounded by two isoparametric tori if the surface \Sigma_{H,C} does not
contain the axis of symmetry or it is dense in the region bounded by a totally
umbilical surface if the surface \Sigma_{H,C} contains the axis of symmetry.
