  In this paper, we characterize all the irreducible Darboux polynomials and
polynomial first integrals of FitzHugh-Nagumo (F-N) system. The method of the
weight homogeneous polynomials and the characteristic curves is widely used to
give a complete classification of Darboux polynomials of a system. However,
this method does not work for F-N system. Here by considering the Darboux
polynomials of an assistant system associated to F-N system, we classified the
invariant algebraic surfaces of F-N system. Our results show that there is no
invariant algebraic surface of F-N system in the biological parameters region.
