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We consider the Dubrovin--Frobenius manifold of rank $2$ whose genusexpansion at a special point controls the enumeration of a higher generageneralization of the Catalan numbers, or, equivalently, the enumeration ofmaps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictlymonotone Hurwitz numbers, or lattice points in the moduli spaces of curves.Liu, Zhang, and Zhou conjectured that the full partition function of thisDubrovin--Frobenius manifold is a tau-function of the extended nonlinearSchr\"odinger hierarchy, an extension of a particular rational reduction of theKadomtsev--Petviashvili hierarchy. We prove a version of their conjecturespecializing the Givental--Milanov method that allows to construct the Hirotaquadratic equations for the partition function, and then deriving from them theLax representation.

Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy