Reciprocal FF-manifolds

We consider the action of a special class of reciprocal transformations on the principal hierarchy associated to a semisimple FF-manifold with compatible flat structure (M,∘,∇,e)(M,∘,∇,e). Under some additional assumptions, the hierarchy obtained applying these reciprocal transformations is also associated to an FF-manifold with compatible flat structure that we call reciprocal FF-manifold. We also consider the special case of bi-flat FF-manifolds (M,∘,∗,∇(1),∇(2),e,E)(M,∘,∗,∇(1),∇(2),e,E) and we study reciprocal transformations preserving flatness of both the connections ∇(1)∇(1) and ∇(2)∇(2) and how they act on corresponding solutions of an augmented Darboux–Egorov system.