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Normal forms for germs of vector fields with quadratic leading part. The polynomial first integral case. (English) Zbl 1321.05134
Summary: We study the problem of formal classification of the vector fields of the form $$\dot{x} = a x^2 + b x y + c y^2 + \dots$$, $$\dot{y} = d x^2 + e x y + f y^2 + \dots$$ using formal changes of the coordinates, but not using the changes of the time. We focus on one special case (which is the most complex one): when the quadratic homogeneous part has a polynomial first integral. In the proofs we avoid complicated calculations. The method we use is effective and it is based on the method introduced in our previous work concerning the Bogdanov-Takens singularity.

##### MSC:
 05C38 Paths and cycles 15A15 Determinants, permanents, traces, other special matrix functions 05A15 Exact enumeration problems, generating functions 15A18 Eigenvalues, singular values, and eigenvectors
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##### References:
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