  We obtain a formula for the number of genus two curves with a fixed complex
structure of a given degree on a del-Pezzo surface that pass through an
appropriate number of generic points of the surface. This is done by extending
the symplectic approach of Aleksey Zinger. This enumerative problem is
expressed as the difference between the symplectic invariant and an
intersection number on the moduli space of rational curves on the surface.
