On B type family of Dubrovin-Frobenius manifolds and their integrable systems

According to D.Zuo and an unpulished work of M.Bertola, there is a two–index series of Dubrovin–Frobenius manifold structures associated to a B type Coxeter group. We study the relations between these structures for the different values of these indices. We show that part of the data of such Dubrovin–Frobenius manifold indexed by $(k,l)$ can be recovered by the $(k+r,l+r)$ Dubrovin–Frobenius manifold. Continuing the program of arXiv:2007.11974 we associate an infinite system of commuting PDEs to these Dubrovin–Frobenius manifolds and show that these PDEs extend the dispersionless BKP hierarchy.