The integrability and linearizability of cubic \(Z_2\)-equivariant systems with two \(1:-q\) resonant saddle points (Q6608694)

The authors investigate the cubic \(Z_2\)-equivariant system \N\begin{align*} \N\dot{x} &= -x/2 - a_{21} y + x^3/2 + a_{21}x^2 y + a_{12} xy^2 + a_{03} y^3,\\\N\dot{y} &= (-q - b_{21}) y + b_{21}x^2 y + b_{12} xy^2 + b_{03} y^3, \N\end{align*} \Nwhere \(q\) is a positive integer. They derive necessary and sufficient conditions on the coefficients for the system to be integrable and linearizable and explicitly calculate the leading ``saddle quantities'' for the normal form system.\N\NIt would have been interesting to explore the derived integrability conditions in the context of the integrability of the normal form equations.