Topological and analytical classification of vector fields with only isochronous centres

We study vector fields on the plane having only isochronous centres. The most familiar examples are isochronous vector fields, they are the real parts of complex polynomial vector fields on C having all their zeroes of centre type. We describe the number N(s) of topologically inequivalent isochronous (singular) foliations that can appear for degree s, up to orientation preserving homeomorphisms. For each s, there exists a real analytic variety I(s) parametrizing the isochronous vector fields of degree s, the group of complex automorphisms of the plane Aut(C) acts on it. Furthermore, if 2≤ s ≤7, then I(s) is a non-singular real analytic variety of dimension s+3, and their number of connected components is bounded by 2N(s). An explicit formula for the residues of the rational 1-form, canonically associated with a complex polynomial vector field with simple zeroes, is given. A collection of residues (i.e. periods) does not characterize an isochronous vector field, even up to complex automorphisms of C. An exact bound for the number of isochronous vector fields, up to Aut(C), having the same collection of residues (periods) is given. We develop several descriptions of the quotient space I(s)/Aut(C) using residues, weighted s-trees and singular flat Riemannian metrics associated with isochronous vector fields.