The present article is the sixth in a series of papers dedicated to two-dimensional cubichomogeneous systems. It considers a case when a homogeneouspolynomial vector in theright-hand part of the system does not have a common factor. Aset of such systems isdivided into classes of linear equivalence, wherein the simplest system being a third-ordernormal form is distinguished on the basis of properly introduced principles. Such a form isdefined by the matrix of its right-hand part coefficients, which is called the canonical form(CF). Each CF has its own arrangement of non-zero elements, their specific normalizationand canonical set of permissible values for the unnormalized elements, which relates the CFto the selected class of equivalence. In addition to classification, each CF is provided with:a) conditions on the coefficients of the initial system, b) non-singular linear substitutionsthat reduce the right-hand part of the system under these conditions to the selected CF,c) obtained values of CF’s unnormalized elements. The proposed classification was primarilycreated to obtain all possible structures of generalized normal forms for systems with CF inthe unperturbed part. The article presents another application of the resulting classificationrelated to finding phase portraits in the Poincare circle forCF