A family of non-darboux-integrable quadratic polynomial differential systems with algebraic solutions of arbitrarily high degree

AbstractWe show that the system xdot = ydot = 2n+2xy+y2 has the algebraic solution h(x, y) = Hn(x)y + 2nHn−1(x), where Hn(x) is the Hermite polynomial of degree n, and the system is not Darboux integrable and has no Darboux integrating factor for any n ϵ N