Complementary 2-forms of Poisson structures

Let (M, P) be a Poisson manifold. A 2-form ω of M such that the Koszul bracket {ω,ω} P = 0 is called a complementary form of P. Every complementary form yields a new Lie algebroid structure of TM, and, under some supplementary hypothesis, the form also defines a Poisson-Nijenhuis structure of M [12]. We give several examples of complementary forms, and new examples of Poisson-Nijenhuis manifolds. The general results are expressed in the framework of Hamiltonian structures [3] and Lie algebroids.