  Generalizing a construction presented in [3], we show that the orbit space of
$B_2$ less the image of coordinate lines under the quotient map is equipped
with two Dubrovin-Frobenius manifold structures which are related respectively
to the defocusing and the focusing nonlinear Schr\"odinger (NLS) equations.
Motivated by this example, we study the case of $B_n$ and we show that the
defocusing case can be generalized to arbitrary $n$ leading to a
Dubrovin-Frobenius manifold structure on the orbit space of the group. The
construction is based on the existence of a non-degenerate and non-constant
invariant bilinear form that plays the role of the Euclidean metric in the
Dubrovin-Saito standard setting. Up to $n=4$ the prepotentials we get coincide
with those associated with constrained KP equations discussed in [20].
