  We consider conformally flat hypersurfaces in four dimensional space forms
with their associated Guichard nets and Lam\'e's system of equations. We show
that the symmetry group of the Lam\'e's system, satisfying Guichard condition,
is given by translations and dilations in the independent variables and
dilations in the dependents variables. We obtain the solutions which are
invariant under the action of the 2-dimensional subgroups of the symmetry
group. For the solutions which are invariant under translations, we obtain the
corresponding conformally flat hypersurfaces and we describe the corresponding
Guichard nets. We show that the coordinate surfaces of the Guichard nets have
constant Gaussian curvature, and the sum of the three curvatures is equal to
zero. Moreover, the Guichard nets are foliated by flat surfaces with constant
mean curvature. We prove that there are solutions of the Lam\'e's system, given
in terms of Jacobi elliptic functions, which are invariant under translations,
that correspond to a new class of conformally flat hypersurfaces.
