Integrable systems, Nijenhuis geometry and Lauricella bi-flat structures

Combining the construction of integrable systems of hydrodynamic type starting from the Frölicher-Nijenhuis bicomplex $(d,d_L)$ associated with a (1,1)-tensor field $L$ with vanishing Nijenhuis torsion with the construction of flat structures starting from integrable systems of hydrodynamic type we define multi-parameter families of bi-flat structures $(\mathrm{\nabla },e,\circ ,{\mathrm{\nabla }}^{\ast },\ast ,E)$ associated with Frölicher-Nijenhuis bicomplexes. We call these structures Lauricella bi-flat structures since in the n-dimensional semisimple case (n-1) flat coordinates of r are Lauricella functions.