A simple way of making a Hamiltonian system into a bi-Hamiltonian one

Given a Poisson structure ( or, equivalently, a Hamiltonian operator) P, we show that its Lie derivative L-tau (P) along a vector field tau defines another Poisson structure, which is automatically compatible with P, if and only if [L-tau(2) ( P), P] = 0, where [., .] is the Schouten bracket. This result yields a new local description for the set of all Poisson structures compatible with a given Poisson structure P of locally constant rank such that dim ker P less than or equal to 1 and leads to a remarkably simple construction of bi-Hamiltonian dynamical systems. A new description for pairs of compatible local Hamiltonian operators of Dubrovin-Novikov type is also presented.