In this paper, we study the center problem for $Z_2$-equivariant quintic vector fields. First of all, for convenience in analysis, the system is simplified
by using some transformations. When the system has two nilpotent points at $(0, ±1)$ with multiplicity three, the first seven Lyapunov constants at the singular points are calculated by applying the inverse integrating factor method.
Then, fifteen center conditions are obtained for the two nilpotent singular
points of the system to be centers, and the sufficiency of the first seven center
conditions are proved. Finally, the first five Lyapunov constants are calculated
at the two nilpotent points $(0, ±1)$ with multiplicity five by using the method
of normal forms, and the center problem of this system is partially solved.
