Normal forms for germs of vector fields with quadratic leading part. The polynomial first integral case

We study the problem of formal classification of the vector fields of the form x˙=ax2+bxy+cy2+… , y˙=dx2+exy+fy2+… 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.