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

The Symmetry Group of Lam\'e's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces