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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>Q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mi>P</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfrac> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mi>E</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> <m:mfrac> <m:mo>∂</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mi>z</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:math> X(z)=Q(z)/P(z)eE(z)∂ /∂ z on the Riemann sphere 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 ∈ 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 ∈ ℰ ( s , r , d ) \ ℰ ( s , r , d ) id with non-trivial isotropy are realized. This yields explicit global normal forms for all X ∈ 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 nontrivial discrete symmetries Γ < Aut( M ), the dictionary describes the correspondence between Γ -invariant tensors on M and tensors on M / Γ .