  In this paper, we consider some cubic near-Hamiltonian systems obtained from
perturbing the symmetric cubic Hamiltonian system with two symmetric singular
points by cubic polynomials. First, following Han [2012] we develop a method to
study the analytical property of the Melnikov function near the origin for
near-Hamiltonian system having the origin as its elementary center or nilpotent
center. Based on the method, a computationally efficient algorithm is
established to systematically compute the coefficients of Melnikov function.
Then, we consider the symmetric singular points and present the conditions for
one of them to be elementary center or nilpotent center. Under the condition
for the singular point to be a center, we obtain the normal form of the
Hamiltonian systems near the center. Moreover, perturbing the symmetric cubic
Hamiltonian systems by cubic polynomials, we consider limit cycles bifurcating
from the center using the algorithm to compute the coefficients of Melnikov
function. Finally, perturbing the symmetric hamiltonian system by symmetric
cubic polynomials, we consider the number of limit cycles near one of the
symmetric centers of the symmetric near-Hamiltonian system, which is same to
that of another center.
