In this paper, we consider the number of zeros of Abelian integral for the system [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are arbitrary polynomials of degree [Formula: see text]. We obtain that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the maximum number of limit cycles bifurcating from the period annulus up to the first order in [Formula: see text]. So, the bounds for [Formula: see text] or [Formula: see text], [Formula: see text], [Formula: see text] are exact.