The transvectants and the integrals for Darboux systems of differential equations

We apply the algebraic theory of invariants of differential equations to integrate the polynomial differential systems $dx/dt=P1_(x,y)+xC(x,y)$, $dy/dt=Q1_(x,y)+yC(x,y)$, where real homogeneous polynomials $P_1$ and $Q_1$ have the first degree and $C(x,y)$ is a real homogeneous polynomial of degree $r\ge 1$. In generic cases the invariant algebraic curves and the first integrals for these systems are constructed. The constructed invariant algebraic curves are expressed by comitants and invariants of investigated systems.