F-Manifolds with Eventual Identities, Bidifferential Calculus and Twisted Lenard-Magri Chains

F-manifolds with eventual identities were introduced by Manin in [17]. In this work, we investigate what these structures entail from the point of view of integrable partial differential equations of hydrodynamic type, especially in terms of the related recurrence relations.After having defined new equivalence relations for connections compatible with respect to the F-product °, namely hydrodynamically almost equivalent and hydrodynamically equivalent connections, we show how these two concepts manifest themselves in several specific situations.In particular, we consider the case of an F-manifold endowed with eventual identity and two hydrodynamically equivalent flat connections and the case of an F-manifold endowed with eventual identity and two almost-hydrodynamically equivalent flat connections. In both cases, we derive generalized recurrence relations for the flows of the associated integrable hierarchy.