  A complete classification of Belyi functions for transforming certain
hypergeometric equations to Heun equations is given. The considered
hypergeometric equations have the local exponent differences 1/k,1/l,1/m that
satisfy k,l,m in N and the hyperbolic condition 1/k+1/l+1/m<1. There are 366
Galois orbits of Belyi functions giving the considered (non-parametric)
hypergeometric-to-Heun pull-back transformations. Their maximal degree is 60,
which is well beyond reach of standard computational methods. To obtain these
Belyi functions, we developed two efficient algorithms that exploit the implied
pull-back transformations.
