Linearizability conditions of time-reversible quartic systems having homogeneous nonlinearities

In this paper we investigate the linearizability problem for the two-dimensional time-reversible complex system ẋ=x+P(x,y),ẏ=−y+Q(x,y), where PP and QQ are homogeneous quartic polynomials. The necessary and sufficient conditions for the linearizability of this system are found. As a corollary, the necessary and sufficient conditions for the origin to be an isochronous center of the planar time-reversible real system u̇=−v+F(u,v),v̇=u+G(u,v), where FF and GG are homogeneous quartic polynomials, are obtained.