All symmetry related invariant manifolds, admitting localized vibrations, for dynamical models on plane square lattice were found by grouptheoretical methods. Discrete breathers were constructed on these manifolds for the model with homogeneous potentials of interparticle interactions and their stability was studied. Nontrivial breather solutions which are not nonlinear normal modes by Rosenberg have been revealed for the above model despite it admits spacetime separation of dynamical variables.
