The Integrability and Linearizability of Cubic \(Z_2\)-Equivariant Systems with Two 1:\(-q\) Resonant Saddle Points

In this article, the integrability and linearizability of a class of cubic \(Z_2\) $Z_2$ -equivariant systems \(\dot{x}=-\frac{1}{2}x-a_{21}y+\frac{1}{2}x^3+a_{21}x^2y+a_{12}xy^2+a_{03} y^3,\, \dot{y}=(-q-b_{21})y+b_{21}x^2y+b_{12}xy^2+b_{03}y^3, \) $\dot{x}=-\frac{1}{2}x-a_{21}y+\frac{1}{2}{x}^3+a_{21}{x}^{2}y+a_{12}x{y}^2+a_{03}{y}^3,\dot{y}=(-q-b_{21})y+b_{21}{x}^{2}y+b_{12}x{y}^2+b_{03}{y}^3,$ are studied. For any positive integer q, we obtain the first three saddle quantities of the above systems by theoretical analysis. Moreover, for any positive integer q,  we derive the necessary and sufficient conditions for the linearizability of the above systems under some assumptions.