Isochronous centers and foci of two-dimensional holomorphic differential systems

Two-dimensional autonomous systems of differential equations of the form $ \dot{x}=-y-P(x,y), \dot{y}=x+Q(x,y), $ where $P$ and $Q$ are holomorphic functions of order greater than or equal two are considered. In this work, necessary and sufficient conditions for the existence of an isochronous center or focus are obtained. These conditions are formulated in terms of commuting differential systems and some normal form.