Complex integrability and linearizability of cubic Z2-equivariant systems with two 1:q resonant singular points

In this paper, complex integrability and linearizability of cubic Z2-equivariant systems with two 1:q resonant singular points are investigated, and the necessary and sufficient conditions on complex integrability and linearizability of the systems with two 1:(−q) resonant saddles are obtained for q=1,2,3,4. Moreover, for general positive integer q, the complex integrability and linearizability conditions are classified, and the sufficiency of the conditions is proved. Further, the linearizability conditions of cubic Z2-equivariant systems with two 1:q resonant node points are also classified.