  This is a complete study of the dynamics of polynomial planar vector fields
whose linear part is a multiple of the identity and whose nonlinear part is a
homogeneous polynomial. It extends previous work by other authors that was
mainly concerned with the existence and number of limit cycles. The general
results are also applied to two classes of examples where the nonlinearities
have degrees 2 and 3, for which we provide a complete set of phase portraits.
