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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**

Saint Petersburg State Technological University of Plant Polymers,

The Faculty of Industrial Heat Power Engineering, Higher Mathematics Department

We consider two-dimensional real autonomous systems of ODE the right-hand parts of which are vector homogeneous polynomials of the third order whose components admit a general multiplier.

In each class of equivalency with respect to linear transformations we distinguish in the right hand part the canonical form, i.e. the simplest polynomial as an unperturbed part of formal or analytical systems which are to be reduced to a generalized normal form.

Conditions on coefficients of a given system and a linear transformation that reduces the system under these conditions to a system with given canonical form are presented.