$GL(2,R)$-orbits of the polynomial sistems of differential equations

In this work we study the orbits of the polynomial systems $\dot x=P(x_1,x_2)$, $\dot x=Q(x_1,x_2)$ by the action of the group of linear transformations $GL(2,R)$. It is shown that there are not polynomial systems with the dimension of $GL$-orbits equal to one and there exist $GL$-orbits of the dimension zero only for linear systems. On the basis of the dimension of $GL$-orbits the classification of polynomial systems with a singular point $O(0,0)$ with real and distinct eigenvalues is obtained. It is proved that on $GL$-orbits of the dimension less than four these systems are Darboux integrable.