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We consider the family  ℰ  ( s ,  r ,  d ) of all singular complex analytic vector fieldsX  (  z  )  =    Q  (  z  )    P  (  z  )e   E  (  z  )∂   ∂  z$X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$   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  &lt; Aut( M ), the dictionary describes the correspondence between  Γ -invariant tensors on  M  and tensors on  M  / Γ .

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