On the number of linear particular integrals of polynomial vector fields

In this paper we consider the ordinary differential equation $P(x,y)dy-Q(x,y)dx=0$ where $P$, $Q$ are relatively prime polynomials of degree, greater than 1. Coefficients of the equations and variables x, y may be complex. We prove that when this equation has an infinite number of linear partial integrals, the polynomials $P$, $Q$ can not be relatively prime. The main result of the paper contains an accurate estimate of the number of different linear particular integrals; estimate of the number of linear integrals when the invariant sets corresponding to line integrals have no points in common; estimate of the number of line integrals in a case where they have a common singular point. The method of proof essentially uses the initial assumption that the polynomials $P$, $Q$ are relatively prime. An example is given that implements proven result.