The authors consider analytic systems of differential equations in the plane whose origin is a nilpotent singularity
where and are natural numbers, , and are quasi-homogeneous vector fields of type and degree with (necessary condition of monodromy). For them, they give an algorithm which provides the Taylor expansion of the return map near the origin. This map is computed by using special polar coordinates, already introduced by Lyapunov, associated to the solution of .
By using their algorithm and other tools described below, they characterize the centers and study the cyclicity of some concrete families of systems of the type given above. These tools are called by the authors conservative-dissipative decomposition of the associated vector field and the following nice result is proved: The nilpotent systems
where are analytic functions defined in a neighborhood of the origin and are integrable analytically in a neighborhood of the origin. This last result extends particular cases considered in the literature, like and others.