Painleve I, Coverings of the Sphere and Belyi Functions
  The theory of poles of solutions of Painleve-I is equivalent to the Nevanlinna problem of constructing a meromorphic function ramified over five points - counting multiplicities - and without critical points. We construct such meromorphic functions as limit of rational ones. In the case of the tritronquee solution these rational functions are Belyi functions. 
Davide Masoero
math-ph math.CA math.MP