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Русская версия

**Vladimir V. Basov**

Universitetsky prospekt, 28,

198504, Peterhof, St. Petersburg, Russia,

Saint-Petersburg State University,

The Faculty of Mathematics and Mechanics,

Differential Equations Department

**Evgenia Fedorova**

Universitetsky prospekt, 28,

198504, Peterhof, St. Petersburg, Russia,

Saint-Petersburg State University,

The Faculty of Mathematics and Mechanics,

Differential Equations Department

Real two-dimensional autonomous systems of ordinary differential equations whose unperturbed parts are vector homogeneous polynomials of the second order, are considered. As to perturbations, they are formal vector power series whose expansions don't contain members of order less then three. A normalization of the unperturbed part of the system is presented. Namely, nineteen classes of equivalence of vector homogeneous polynomials with respect to any linear invertible transformations, are founded. Each class is represented by a canonical form, i.e. by a polynomial of the definite structure having the maximal number of zero coefficients. Generalized normal forms are explicitly given for five types of systems whose unperturbed parts are of degenerate canonical forms. These normal forms can be obtained by almost identical formal transformations.