On some involution theorems on twofold Poisson manifolds

We point out some involution theorems which are consequences of the existence of two compatible Poisson structures on a manifold. Using a theorem of Lichnerowicz on local triviality of the Schouten-Nijenhuis cohomology, we show that local exactness of the second Poisson structure with respect to the ground one is equivalent to involutivity of the algebra of invariant functions of the ground structure. Then an involution theorem of Mishchenko and Fomenko is given, founded on global exactness of the second structure. Finally a generalization of a recurrence operator is given to obtain a set of traces which are in involution.