  We compute the number of coverings of ${\mathbb{C}}P^1\setminus\{0, 1,
\infty\}$ with a given monodromy type over $\infty$ 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.
