  A generating function of the single Hurwitz numbers of the Riemann sphere
$\mathbb{CP}^1$ is a tau function of the lattice KP hierarchy. The associated
Lax operator $L$ turns out to be expressed as $L = e^{\mathfrak{L}}$, where
$\mathfrak{L}$ is a difference-differential operator of the form $\mathfrak{L}
= \partial_s - ve^{-\partial_s}$. $\mathfrak{L}$ satisfies a set of Lax
equations that form a continuum version of the Bogoyavlensky-Itoh (aka hungry
Lotka-Volterra) hierarchies. Emergence of this underlying integrable structure
is further explained in the language of generalized string equations for the
Lax and Orlov-Schulman operators of the 2D Toda hierarchy. This leads to
logarithmic string equations, which are confirmed with the help of a
factorization problem of operators.
