Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

We consider the family E (s, r, d) of all singular complex analytic vector fields X(z) = Q(z)/P(z) e(E(z))partial derivative/partial derivative z on the Riemann sphere (C) over cap, where Q, P, E are polynomials with deg Q = s, deg P = r and deg E = d >= 1. Using the pullback action of the affine group Aut(C) and the divisors for X, we calculate the isotropy groups Aut(C)(X) of discrete symmetries for X is an element of E (s, r, d). The subfamily E (s, r, d)(id) of those X with trivial isotropy group in Aut(C) is endowed with a holomorphic trivial principal Aut(C)-bundle structure. A necessary and sufficient arithmetic condition on s, r, d ensuring the equality E (s, r, d) = E (s, r, d)(id) is presented. Moreover, those X is an element of E (s, r, d) \ E (s, r, d)(id) with non-trivial isotropy are realized. This yields explicit global normal forms for all X is an element of E (s, r, d). A natural dictionary between analytic tensors, vector fields, 1-forms, orientable quadratic differentials and functions on Riemann surfaces M is extended as follows. In the presence of non-trivial discrete symmetries Gamma < Aut(M), the dictionary describes the correspondence between G-invariant tensors on M and tensors on M/Gamma.