Integrable hierarchies, Frolicher-Nijenhuis bicomplexes and Lauricella bi-flat F-manifolds

Given the Frolicher-Nijenhuis bicomplex (d,dL) associated with a (1,1) -tensor field L with vanishing Nijenhuis torsion, we define a multi-parameter family of bi-flat structures ( backward difference ,e,circle, backward difference *,*,E) . This result is obtained by combining the construction of integrable hierarchies of hydrodynamic type starting from Frolicher-Nijenhuis bicomplexes with the construction of flat F-manifold structures from integrable systems of hydrodynamic type. By construction L is the operator of multiplication by the Euler vector field E and the number of parameters coincides with the number of Jordan blocks appearing in its Jordan normal form. We call these structures Lauricella bi-flat structures since in the n-dimensional semisimple case (n-1) flat coordinates of backward difference are Lauricella functions. The (1,1) -tensor fields defining the corresponding integrable hierarchies have a similar block diagonal structure.