The special setQ ofWightman's functionals is considered. All the pure functionals that belong toQ have the strong spectral property and the single vacuum. All other functionals fromQ may be represented by integrals over (weak) compact sets of pure functionals fromQ. The Borchers structure theorem is proved for this set. The conditions of the unitary equivalence of the representations and their similarity are given. These results are applied for the analysis of the symmetry problem. In particular we formulate the necessary and sufficient condition for the given group of automorphisms to be unitarily implemented.