  We study the structure of the Goulden-Jackson-Vakil formula that relates
Hurwitz numbers to some conjectural "intersection numbers" on a conjectural
family of varieties $X_{g,n}$ of dimension $4g-3+n$. We give explicit formulas
for the properly arranged generating function for these "intersection numbers",
and prove that it satisfies Hirota equations. This generalizes and
substantially simplifies our earlier results with Zvonkine in
arXiv:math/0602457.
