Two-dimensional homogeneous cubic systems: Classification and normal forms–III

This article is the third in a series of works devoted to two-dimensional homogeneous cubic systems. It considers the case where the homogeneous polynomial vector on the right-hand side of the system has a quadratic common factor with real zeros. The set of such systems is divided into classes of linear equivalence, in each of which a simplest system being a third-order normal form is distinguished on the basis of appropriately introduced structural and normalization principles. In fact, this normal form is determined by the coefficient matrix of the right-hand side, which is called a canonical form (CF). Each CF is characterized by an arrangement of nonzero elements, their specific normalization, and a canonical set of admissible values of the unnormalized elements, which ensures that the given CF belongs to a certain equivalence class. In addition, for each CF, (a) conditions on the coefficients of the initial system are obtained, (b) nonsingular linear substitutions reducing the right-hand side of a system satisfying these conditions to a given CF are specified, and (c) the values of the unnormalized elements of the CF thus obtained are given.