Virasoro constraints and topological recursion for Grothendieck's dessin counting

We compute the number of coverings of ${$\mathbb{C}$}P^1\setminus{0, 1, \infty}$withagivenmonodromytypeover$\infty$ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchy and satisfies a topological recursion in the sense of Eynard-Orantin.