A class of integrable polynomial vector fields

We study the integrability of two-dimensional autonomous systems in the plane of the form ${\displaystyle \stackrel{.}{x}=-y+X_s(x,y)}$, ${\displaystyle \stackrel{.}{y}=x+Y_s(x,y)}$, where X_s(x,y) and Y_s(x,y) are homogeneous polynomials of degree s with s≥2. First, we give a method for finding polynomial particular solutions and next we characterize a class of integrable systems which have a null divergence factor given by a quadratic polynomial in the variable ${\displaystyle {(x^2+y^2)}^{\frac{s}{2}-1}}$ with coefficients being functions of tan^{−1}(y/x).