A note on analytic integrability of planar vector fields

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems (x) over dot = (partial derivative h/partial derivative y (x, y)K(h, y(n)) + y(n-1)Psi(h, y(n))) xi(x, y), (y) over dot = -partial derivative h/partial derivative x(x, y)K(h, y(n))xi(x, y), where h, K, Psi and xi are analytic functions defined in the neighbourhood of O with K(O) not equal 0 or Psi(0) not equal 0 and n >= 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.