Centre families in two-dimensional complex holomorphic dynamical systems

We consider a two-dimensional complex holomorphic dynamical system. In particular, we use the singular point theory of C. H. Briot and J. C. Bouquet to establish the existence of complex holomorphic invariant manifolds of the system in the neighbourhood of an equilibrium point with two purely imaginary eigenvalues. Consequently, this enables us to establish the existence of isochronous centre families in the neighbourhood of the equilibrium point. The results are exhibited by application to the complex Takens-Bogdanov system.