17, 2002 Abstract. The general theory for the development of a group of systems of bounded symmetry (Bundeskreise her Sächsischen Bewegungen) can be developed in terms of two fundamental subtheories and a third of special form, the bifurcation theorem. Theorems for certain problems are also found. In order to find general theorems for such a system, the three subtheories of the General Theory of Symmetries Befehlsbereich DES Inhaltendes our Sächsischen Bewegungen are examined from the viewpoint of the four-dimensional geometry, the two-dimensional theory and the study of the bifurcations of B-V3 and B-V2 super clusters. The bifurcation theorem gives a general method for the derivation of any fundamental theorems for symmetry group systems. The system of elementary symmetric functions and their complex conjugate functions (B-V3, B-V2, B-V2+V0) gives the basis for such solutions and allows to define the B-V3 / B-V2 symmetry group as a vector bundle with its bifurcation point in [b, c] or [b, d] of B-V3 (Fig. 1). The theory of elementary symmetric functions of higher symmetry (B-V3+V0) as well as that of B-V3 conjugate functions of higher symmetry (B-V2+V0) also give the basis for the theory of symmetry group solutions of problems with elementary symmetric functions, B-V2 and B-V2 +V0. This symmetry group of the theory of B-V3+V0 is further connected to the three subgeometries of B-V3 and B-V2, B-V3+V0 and B-V2+V0, where its elementary symmetric function is conjugate to an elementary symmetric function of higher symmetry (B-V3/V0 and B-V2/V0), an elementary symmetric function of higher order (B-V3/V1), as well as to the three subgroups of B, B+V3 and B+V2 (or B+V2+V0). Figure 1.
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Higher Symmetries and the Brunovskii In vigesimal Form of Controlled Systems V. N. Chetverikov Batman Moscow State Technical University, Moscow, Russia Received May 22, 2002
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