Generalized Frobenius Structures on Hurwitz Spaces, Their Open Extensions and Associated Principal Hierarchies

We construct generalized Frobenius structures on Hurwitz spaces of arbitrary genus using extended primary differentials. Through contour integrals involving the Cauchy kernel, we derive explicit dispersionless Lax representations for the principal hierarchies of these structures and their almost-dual counterparts. We also construct solutions to the open WDVV equations for both the original and almost-dual structures, and derive explicit principal hierarchies for the resulting rank-one extensions. Finally, we show that these constructions admit Shramchenko deformations.