Simultaneous integrability and non-linearizability at arbitrary double weak saddles and sole weak focus of a cubic Liénard system

In this paper, we study the problems of simultaneous integrability and non-linearizability of arbitrary double weak saddles and sole weak focus for a planar cubic Liénard system with cubic damping and restoring force. First of all, we discriminate the equilibria type: two of saddle type and one of focus-center type. Next, via computing saddle values and focus values, we deduce the integrability condition under which the system is simultaneously integrable at double weak saddles and sole weak focus. The sufficiency of the condition is proved as well. Lastly, we come to the conclusion that the system is not linearizable at the three equilibria by means of computing linearizability constants.