Linearizability conditions of time-reversible cubic systems

In this paper we present the necessary and sufficient conditions for linearizability of the planar time-reversible cubic complex system x ˙ = x + P ( x , y ) , y ˙ = − y + Q ( x , y ) . From these conditions, the necessary and sufficient conditions for the origin to be an isochronous center of the time-reversible cubic real system u ˙ = − v + F ( u , v ) , v ˙ = u + G ( u , v ) can be obtained. Thus, the isochronous center problem of time-reversible cubic systems is solved completely.