Counting Integral Lam\'e Equations by Means of Dessins d'Enfants

We obtain an explicit formula for the number of Lam'e equations (modulo scalar equivalence) with index $n$ and projective monodromy group of order $2N$, for given $n\in \mathbb{Z}$ and $N\in \mathbb{N}$. This is done by performing the combinatorics of the `dessins d’enfants’ associated to the Belyi covers which transform hypergeometric equations into Lam'e equations by pull-back.