Regular F-manifolds with eventual identities

Given an F-manifold one may construct a dual multiplication (generalizing the idea of an almost-dual Frobenius manifold introduced by Dubrovin) using a so-called eventual identity, the definition of which ensures that the dual object is also an F-manifold. In this paper we solve the equations for an eventual identity for a regular (so non-semi-simple) F-manifold and construct a dual coordinate system in which dual multiplication is preserved. As an application, families of Nijenhuis operators are constructed.