Hurwitz numbers and integrable hierarchy of Volterra type

A generating function of the single Hurwitz numbers of the Riemann sphere ${\mathbb{C}\mathbb{P}}^1$ 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 $\mathfrak{L} = \partial_s - ve^{-\partial_s}$\mathrm{.}$\mathfrak{L}$ 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.