Generalized normal forms of the systems of ordinary differential equations with a quasi-homogeneous polynomial $(\alpha x_1^2+x_2, x_1 x_2)$ in the unperturbed part

In this paper a study on constructive construction of the generalized normal forms (GNF) is continued. The planar real-analytical at the origin system is considered. Its unperturbed part forms a first degree quasi-homogeneous first degree polynomial $(\alpha x_1^2+x_2, x_1 x_2)$ of type $(1, 2)$ where parameter $\alpha \in (-1/2, 0) \cup (0, 1/2]$. For given value of this polynomial is a canonical form, that is an element of a class of equivalence relative to quasi-homogeneous substitutions of zero order into which any first order quasi-homogeneous polynomial of type $(1, 2)$ is divided in accordance with the chosen structural principles due to it only making sense to reduce the systems with the various canonical forms in their unperturbed part to GNF. Based on the constructive method of resonance equations and sets, the resonance equations are derived. Perturbations of the acquired system satisfies these equations if an almost identity quasi-homogeneous substitution in the given system is applied. Their validity guarantees formal equivalence of the systems. Besides, resonance sets of coefficients are specified that allows to get all possible GNF structures and prove reducibility of the given system to a GNF with any of specified structures. In addition, some examples of characteristic GNFs are provided including that with the parameter leading to appearance of an additional resonance equation and the second nonzero coefficient of the appropriate orders in GNFs.