  We construct a symplectic realization and a bi-hamiltonian formulation of a
3-dimensional system whose solution are the Jacobi elliptic functions. We
generalize this system and the related Poisson brackets to higher dimensions.
These more general systems are parametrized by lines in projective space. For
these rank 2 Poisson brackets the Jacobi identity is satisfied only when the
Pl\" ucker relations hold. Two of these Poisson brackets are compatible if and
only if the corresponding lines in projective space intersect. We present
several examples of such systems.
