On Integrability of Generalized Veronese Curves of Distributions

Given a 1-parameter family of 1-forms γ(t) = γ0 + tγ1 + . . .+ t nγn, consider the condition dγ(t) ∧ γ(t) = 0 (of integrability for the annihilated by γ(t) distribution w(t)). We prove that in order that this condition is satisfied for any t it is sufficient that it is satisfied for N = n+3 different values of t (the corresponding implication for N = 2n+1 is obvious). In fact we give a stronger result dealing with distributions of higher codimension. This result is related to the so-called Veronese webs and can be applied in the theory of bihamiltonian structures.