Classifying (almost)-Belyi maps with five exceptional points

Abstract We classify all rational functions f : P 1 → P 1 whose branching pattern above 0 , 1 , ∞ satisfy a certain regularity condition with precisely d = 5 exceptions. This work is motivated by solving second order linear differential equations, with d = 5 true singularities, in terms of hypergeometric functions. A similar problem was solved for d = 4 in Vidunas and Filipuk (2013).