In this paper we relate the geometric Poisson brackets on the Grassmannian of
2-planes in R^4 and on the (2,2) Moebius sphere. We show that, when written in
terms of local moving frames, the geometric Poisson bracket on the Moebius
sphere does not restrict to the space of differential invariants of Schwarzian
type. But when the concept of conformal natural frame is transported from the
conformal sphere into the Grassmannian, and the Poisson bracket is written in
terms of the Grassmannian natural frame, it restricts and results into either a
decoupled system or a complexly coupled system of KdV equations, depending on
the character of the invariants. We also show that the biHamiltonian
Grassmannian geometric brackets are equivalent to the non-commutative KdV
biHamiltonian structure. Both integrable systems and Hamiltonian structure can
be brought back to the conformal sphere.Comment: 33 page
