  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 $[\cdot,\cdot]$ is the Schouten bracket.
We further prove that if $\dim\ker P\leq 1$ and $P$ is of locally constant
rank, then all Poisson structures compatible with a given Poisson structure $P$
on a finite-dimensional manifold $M$ are locally of the form $L_{\tau}(P)$,
where $\tau$ is a local vector field such that
$L_{\tau}^2(P)=L_{\tilde\tau}(P)$ for some other local vector field
$\tilde\tau$. This leads to a remarkably simple construction of bi-Hamiltonian
dynamical systems. We also present a generalization of these results to the
infinite-dimensional case. In particular, we provide a new description for
pencils of compatible local Hamiltonian operators of Dubrovin--Novikov type and
associated bi-Hamiltonian systems of hydrodynamic type.
  Key words: compatible Poisson structures, Hamiltonian operators,
bi-Hamiltonian systems (= bihamiltonian systems), integrability, Schouten
bracket, master symmetry, Lichnerowicz--Poisson cohomology, hydrodynamic type
systems.
  MSC 2000: Primary: 37K10; Secondary: 37K05, 37J35
