A class of quintic \(Z_2\) -equivariant systems possessing four invariant straight lines, with singularities located at \((\pm 1, 0)\) are studied. At first, when \((\pm 1, 0)\) are weak foci, the first six Lyapunov constants are computed by using the formal series method. The necessary conditions for integrability are formulated by equating these constants to zero. Then, when \((\pm 1, 0)\) are nilpotent singular points, the first five quasi-Lyapunov constants are obtained using the inverse integrating method. The requisite integrability conditions emerge from nullifying these constants. Consequently, when \((\pm 1, 0)\) are weak saddles, the first six saddle quantities are calculated. The necessary integrability conditions are acquired by setting these quantities to zero. Finally, the sufficiency of all the above conditions is rigorously proved through the application of the Symmetry Principle, the technique of constructing first integrals, and the method of deriving integrating factors.
