Higher-Order Strong Isochronism of Cauchy–Riemann Systems with Holomorphic Perturbations of a Linear Center

We consider systems of the form ẋ=-y-P(x,y) , ẏ=x+Q(x,y) , where P and Q are functions holomorphic at the origin whose power series expansions in x and y do not contain free and linear terms and which satisfy the Cauchy–Riemann conditions. The maximum orders of the strong general isochronism and the strong partial (with given initial polar angle) isochronism are determined for systems in a fairly broad subclass of such systems.